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Methods 
for 

Sensory 
Evaluation 

of 
Food 



SEP 2 4 1973 

Agriculture 
Canada 



I* 



;vised 



630.4 

C212 

P1284 

1970 

(1973print) 

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PUBLICATION 1284 



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PUBLICATION 1284 1970 



METHODS FOR SENSORY EVALUATION OF FOOD 



ELIZABETH LARMOND 
Food Research Institute, Central Experimental Farm, Ottawa 



CANADA DEPARTMENT OF AGRICULTURE 



Copies of this publication may be obtained from 

INFORMATION DIVISION 

CANADA DEPARTMENT OF AGRICULTURE 

OTTAWA 

K1A0C7 

©INFORMATION CANADA. OTTAWA. 1973 

Printed 1967 
Revised 1970 
Reprinted 1971, 1973 

Code No.: 3M-3651 3-9:73 
Cat. No.: A53-1284 



CONTENTS 

Introduction 3 

Types of Tests 5 

Samples and Their Preparation 5 

Panelists 7 

Testing Conditions 8 

Questionnaires 10 

Design of Experiments and Methods of Analyzing Data .... 13 

Consumer Testing 14 

Appendix I - Sample Questionnaires and Examples of 
Analyses 

Triangle Test Difference Analysis 15 

Duo-Trio Test Difference Analysis 17 

Multiple Comparison Difference Analysis 19 

Ranking Difference Analysis 24 

Scoring Difference Test 27 

Paired Comparison Difference Test 31 

Hedonic Scale Scoring 36 

Paired Comparison Preference 37 

Ranking Preference 38 

Appendix 1 1 - Statistical Chart 1 39 

Statistical Chart 2 - F Distribution 40 

Statistical Chart 3 - Studentized Ranges, 5 percent ... 42 

Statistical Chart 4 - Studentized Ranges, 1 percent ... 43 

Statistical Chart 5 - Scores for Ranked Data 44 

Appendix IN - Notes on Introductory Statistics by Andres 

Petrasovits 45 

References 55 

Additional Sources of Information 56 



Digitized by the Internet Archive 

in 2012 with funding from 

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http://www.archive.org/details/methodsforsensorOOIarm 



INTRODUCTION 

A sensory evaluation is made by the senses of taste, smell, and touch 
when food is eaten. The complex sensation that results from the interaction 
of our senses is used to measure food quality in programs for quality con- 
trol and new product development. This evaluation may be carried out by 
one person or by several hundred. 

The first and simplest form of sensory evaluation is made at the bench 
by the research worker who develops the new food products. He relies on 
his own evaluation to determine gross differences in products. Sensory 
evaluation is conducted in a more formal manner by laboratory and consumer 
panels. 

Most aspects of quality can be measured only by sensory panels, 
although advances are being made in the development of objective tests 
that measure individual quality factors. Instruments that measure texture 
are probably the best known. Some examples are the L.E.E. -Kramer shear 
press and the Warner-Bratzler shearing device. Gas chromatography and 
mass spectrometry enable odor to be measured to a limited extent. The 
color of foods can be accurately measured by tristimulus colorimetry. As 
new instruments are developed to measure quality, sensory evaluation will 
be used to prove and standardize new objective tests. 

When people are used as a measuring instrument, it is necessary to 
rigidly control all testing methods and conditions to overcome errors caused 
by psychological factors. "Error" is not synonymous with mistakes, but 
may include all kinds of extraneous influences. The physical and mental 
condition of the panelist and the influence of the testing environment affect 
sensory tests. For example, some people may have more flavor acuity in 
the morning, others in the afternoon. Even the weather can influence the 
disposition of panelists. 

Small panels are used to test the palatability of foods. They may also 
be used in preliminary acceptance testing. Laboratory panels may be used 
to determine: 

the best processing procedures, 

suitable varieties of raw material, 

preferable cooking and processing temperatures, 

effect of substituting one ingredient for another, 

best storage procedures, 

effect of insecticides and fertilizers on flavor of foods, 

effect of animal feeds on flavor and keeping quality of meat, 

optimum size of pieces and their importance, 

effect of color on acceptability of foods, 

suitable recipes for the use of new products, 

comparison with competitors' products. 



The author is grateful to Mr. A. Petrasovits, Statistical Research 
Service, Canada Department of Agriculture, who wrote Appendix III. 
Statistical Chart 1 has been reprinted with permission from Wallerstein 
Laboratory (Bengtsson, K. 1953. Wallerstein Lab. Commun. 16 (No. 54): 
231—251.). Thanks are due to the Literary Executor of the late Sir Ronald 
A. Fisher, F.R.S., Cambridge, to Dr. Frank Yates, F.R.S. Rothamstead, 
and to Messrs. Oliver and Boyd Ltd., Edinburgh, for permission to reprint 
in part Tables V (Statistical Chart 2) and XX (Statistical Chart 5) from 
their book Statistical Tables for Biological, Agricultural and Medical 
Research. Thanks are also due to Dr. Malcolm Turner for permission to 
reprint "Multiple Range and Multiple F Tests" (Statistical Charts 3 and 4) 
by D. B. Duncan from Biometrics, Volume II, 1955. 



4 



TYPES OF TESTS 

The two types of tests are difference tests and preference tests. 

Difference Tests 

In difference tests the members of the panel are merely asked if a 
difference exists between two or more samples. Individual likes and dis- 
likes are disregarded and each panelist is advised to be objective in his 
evaluation. He may not like a particular product, but he should be taught 
what constitutes good and poor quality and try to evaluate it on the basis 
of the instructions received. He is acting as a quality measuring instru- 
ment. 

Preference Tests 

Preference or acceptance tests determine representative population 
preferences, and need many people on the panel. The total scores from 
trained panels can be used to predict preference scores obtained from 
panels of 100 to 160 untrained persons (19). Some tests are conducted on 
a national scale by firms specializing in this form of testing. Even after 
extensive tests, there is no assurance that the results will apply to the 
total population. 

SAMPLES AND THEIR PREPARATION 

Panel members are usually influenced by all the characteristics of the 
test material. Therefore, test samples should be prepared and served as 
uniformly as possible (6). 

Information About Samples 

As little information as possible about the test should be given to the 
panelists, since this information may influence results. It has been found 
(13) that if information given to the panelists has meaning within the terms 
of their experience it will influence their responses; panelists will taste 
what they expect to taste. When panelists were told that high-quality raw 
products were used, the panel preference was high, whereas the knowledge 
that low-quality raw products were used lowered the panel preference. The 
information that the food had been treated with unspecified chemicals and 
exposed to unspecified sterilizing rays did not alter panel findings. 

Temperature of Samples 

The samples must be of uniform temperature. Therefore the mechanical 
problems of serving foods at a constant and uniform temperature should be 



carefully considered. The temperature at which food is usually eaten is 
recommended for samples. However, taste buds are less sensitive to very 
high or very low temperatures, which impair full flavor perception. 

Sample Uniformity 

To measure flavor differences in products of large unit size, such as 
canned peach halves, slice the large units into smaller pieces and care- 
fully mix them to obtain a more uniform sample. To test the quality of canned 
juice, open several cans and mix all the juice together before preparing 
individual test samples. 

Coding 

Samples should be coded in such a manner that the judges cannot 
distinguish the samples by the code or be influenced by code bias. For 
example, if the samples are numbered 1, 2, 3, or lettered A, B, C, a coding 
bias could be caused because people associate 1 or A with "first" or 
"best" and might tend to score this sample higher. A set of three-digit 
random numbers should be assigned to each sample so that the panelists 
will receive samples coded differently (17). 

Number of Samples 

To determine the number of samples to be presented at one testing 
session (12) consider the following: 

The nature of the product being tested — No more than six samples of 
ice cream should be evaluated because of the temperature of the product. 

The itensity and complexity of the sensory property being judged — 
It has been found (18) that with mild products such as green beans and 
canned peaches up to 20 samples may be tasted with no decrease in the 
taster's ability to discriminate. 

The experience of the taster — A professional tea, wine, or coffee 
taster can evaluate hundreds of samples in one day. 

The amount of time and commodity available. 

Order of Presentation 

The order in which the samples are presented to the judges may also 
influence results. Studies on the effect of sample sequence on food prefer- 
ence (8) have shown position effect (the later samples were rated lower); 
contrast effect (serving "good" samples first lowered the ratings for "poor" 
samples); and convergence effect (tendency to make similar responses to 
successive stimuli). Contrast and convergence effects were shown to be 
independent of position effect. Random presentation to the panelists 
equalized these effects. 




Figure 1. Preparation of samples. 

Figure 2. Presentation of samples through hatch. 

PANELISTS 



For economical reasons choose panel members from all available person- 
nel, including office, plant, and research staff. It should be considered a 
part of work routine for personnel in the food industry to serve as panelists. 
However, no one should be asked to evaluate foods to which he objects. 
Persons concerned with the test product (product development and produc- 
tion) and those who prepare the samples for testing should not be included 
on the panel. 

Panel members should be in good health and should excuse themselves 
if they are suffering from a cold. Nonsmokers and smokers have been found 
equally useful for panels. It is inadvisable for smokers to smoke within one 
to two hours before a test (3). Heavy smokers, people who smoke one or 
more packs of cigarettes per day, have been found to be generally less 
sensitive than nonsmokers (2). There are exceptions however. No correlation 
exists between age and sensitivity. A certain amount of sensitivity is 
required, but this seems not as important as experience (1). A person of 
average sensitivity, a high degree of personal integrity, ability to concen- 
trate, intellectual curiosity, and willingness to spend time in evaluation 
may do a better job than a careless person with extreme acuity of taste and 
smell. 



To keep the panelists interested in the work that is being done, show 

them the results when each series of tests is completed. The importance 

of the work can be shown by running the tests in a controlled, efficient 
manner. 

Selection and Training 

Panelists should be selected for their ability to detect differences. 
People who do well with some products often do poorly on others. A taster 
is seldom equally proficient in tasting all foods. 

Researchers disagree on the value of training panelists. It has been 
stated (17) that screening tests could be used to choose panelists who are 
capable of detecting differences, and actual training would be unnecessary. 
At the Operational Research Group of Cadbury Bros. Ltd., in Bournville, 
England, panelists with sufficiently discriminating palates are selected on 
the basis of a series of triangular tests with typical confectionery materials. 
The selected panelists are given 10 weeks training on special training 
panels and finally they attend the formal sessions of a tasting panel for 
4 weeks before their assessments are actually used (15). Certainly a specific 
panel for the product and method being tested is more useful than a general- 
purpose panel. Panelists must "be familiar with the product and must know 
what constitutes good quality in the product. Preliminary sessions will 
help to clarify the meaning of descriptive terms. To be of any real use terms 
should be referable to specific objective standards (23). 

Number of Panelists 

Since there are many sources of variation in sensory tests, the more 
tasters on the panel, the more likely it is that the individual variations will 
balance. Four panelists is probably the minimum number (11), although most 
researchers think there should be eight or ten (10). A small panel of high 
sensitivity and ability to differentiate may be preferable to a large panel of 
less sensitivity. 

TESTING CONDITIONS 

Testing Area 

The room where the tests are run may be simple or elaborate, but the 
panelists must be independent of each other, in separate booths or in parti- 
tioned sections at a large table. The panelists must be free from distractions 
such as noise. 

If possible, the room should be odor free and separate from, though 
adjacent to, the area where the sample is prepared. Air-conditioning is 
useful in the testing area. The walls should be off-white or a light neutral 
gray so that sample color will not be altered. 



8 




Figure 3. Taste panel area. 

Lighting 

Uniform lighting is essential. Natural white fluorescent lights are more 
suitable than cool white lights. Red fluorescent lights are used in the test- 
ing area of the Food Research Institute, Ottawa, to hide obvious color 
differences in samples whose flavor is being evaluated. 

Testing Schedule 

The time of day that tests are run influences results. Although this 
cannot be controlled if the number of tests is large, late morning and 
midafternoon have been found to be the best times for testing. Since a 
panelist's eating habits affect test results, it is desirable that no tests be 
performed in the period from 1 hour before a meal to 2 hours after a meal. 

Containers 

The samples to be tested should be presented to the panelists in clean, 
odorless, and tasteless containers. 



Tasting Procedures 

It is generally agreed that whether a panelist swallows the sample or 
spits it out the result of the test is unchanged. However, the panelist should 
be instructed to use the same method with each sample in each test. 



Use crackers, white bread, celery, apples, or water to remove all traces 
of flavor from the mouth between tasting samples of certain foods. If water 
is used it should be at room temperature, as cold water reduces the effi- 
ciency of the taste buds. 



QUESTIONNAIRES 

The simplest type of questionnaire has been found to be the most effi- 
cient. Elaborate questionnaires divert the attention of the panelist and 
complicate interpretations. No blanks should be put in the questionnaire 
except those that apply to the panelist. The person analyzing the test 
results should use separate summary sheets. 

A new questionnaire should be prepared if the method or objective of 
the test changes. The increased reliability of the test results is worth the 
time and effort needed to make up new questionnaires. 

Questionnaires for some commonly used tests follow. Sample question- 
naires with examples of their analyses are shown in Appendix I. 




Figure 4. Panelist with samples and questionnaire. 



10 









Figures 5—12. Examples of trays prepared for the following tests: 5, triangle test; 
6, duo»trio test; 7, paired comparison test; 8, ranking for difference; 9, multiple 
comparison; 10, scoring for difference; 11, scoring for preference; 12, ranking 
for preference. 



11 



Difference Tests 

The triangle test — In the triangle test, three coded samples are pre- 
sented to the panelist. He is told that two samples are identical and he is 
asked to indicate the odd one. See sample questionnaire on page 15. 

The duo-trio test — In the duo-trio test, three samples are presented to 
the taster, one is labeled R (reference) and the other two are coded. One 
coded sample is identical with R and the other is different. The panelist is 
asked to identify the odd sample. See sample questionnaire on page 17. 

Paired comparison test — In the paired comparison test, a pair of coded 
samples that represent the standard or control and an experimental treatment 
are presented to the panelist, who is asked to indicate which sample has the 
greater or lesser degree of intensity of a specified characteristic — such as 
sweetness and hardness. If more than two treatments are being considered, 
each treatment is compared with every other in the series. The design be- 
comes somewhat cumbersome if many treatments are compared. 

An example of a paired comparison test is given on page 31. 

In most instances the value of a paired comparison can be increased 
by a panelist's report on the extent of the difference found (20). 

Paired testing is generally used to compare new with old procedures 
and in quality control. When evidence from paired comparison experiments 
is reported, do not conclude without further facts that a less preferred 
treatment is of poor quality. The panelist should be asked to rate the quality 
of each treatment as good, fair, or poor. 

Ranking — The panelist is asked to rank several coded samples accord- 
ing to the intensity of some particular characteristic. See sample question- 
naire on page 24. 

Multiple comparison — In multiple comparison tests, a known reference 
or standard sample is labeled R and presented to the panelist with several 
coded samples. The panelist is asked to score the coded samples in com- 
parison with the reference sample. See sample questionnaire on page 19. 

Scoring — Coded samples are evaluated by the panelist who records his 
reactions on a descriptive graduated scale. These scores are given numerical 
values by the person who analyzes the results. See sample questionnaires 
on pages 27 and 30. 

The flavor-profile method — The flavor-profile procedure was developed 
by Arthur D. Little Incorporated. A small laboratory panel of six or eight 
people who have been trained in the method measure the flavor profile of 
food products. Descriptive words and numbers, with identical meaning to 
each panel member, are used to show the relative strength of each note on a 
suitable scale. With this method it is possible to determine small degrees 



12 



of difference between two samples, the degree of blending, degrees of sim- 
ilarity, and overall impression of the product. Considerable knowledge of 
flavor is required for the interpretation of flavor-profile results, since they 
cannot be analyzed statistically. The flavor-profile method requires great 
skill, extensive education in odor and flavor sensations, and keen interest 
and intelligence on the part of the panelist. This technique has been re- 
viewed in detail by Sjostrom (21) and Caul (4). 

Dilution tests — Dilution tests involve the determination of the identi- 
fication threshold for the material under study. The flavor of a product is 
described in terms of the percentage of dilution or as a ratio that reflects 
the actual amount of odor or flavor detected. This method requires suitable 
standards for comparison and for dilution of the test material and is limited 
to foods that can be made homogeneous without affecting flavor (24). 

Preference Tests 

Paired comparison — The paired comparison test used in preference 
testing is similar to that used for difference testing. When testing prefer- 
ences, the panelist is asked which sample he prefers and the degree of 
preference. See sample questionnaire, page 37. 

Scoring — Many different types of scales have been developed to try to 
determine a degree of like or dislike for a food. These scales may be worded 
"excellent, " "very good," "good," "poor," or in other similar manner. 
However, the preference scale that has probably received the most attention 
in the past 10 years is the nine-point hedonic scale developed at the Quar- 
termaster Food and Container Institute of the United States. Much time and 
effort was expended to determine which words best express a person's like 
or dislike of a food. This hedonic scale is the result of these investigations 
(14). See sample questionnaire on page 36. 

Ranking — Ranking follows the same procedure as difference testing 
(p. 12) except that when used as a preference test the questionnaire is 
worded so that the panelist will indicate his order of preference for the 
samples. See sample questionnaire on page 38. 



DESIGN OF EXPERIMENTS AND METHODS OF ANALYZING DATA 

The accuracy of sensory evaluation tests on food and the reliance that 
can be placed on their results depend on standardization of testing conditions 
and use of statistical methods of experimental design and analysis (22). 

Plan the experiment in advance so that a simple mathematical model 
may be applied to the analysis. Many simple mathematical models depend on 
the independent nature of the data. Experimental design was introduced to 



13 



develop this independence and often makes a- simple mathematical model 
applicable. Experimental design makes the test efficient and saves time 
and material. An excellent reference for experimental design has been 
written by Cochran and Cox (5). Application of an experimental design 
should be approved by a statistician. 

Random choice of samples and order of presentation help eliminate error. 
Replication of tests will strengthen the results. 

A discussion of the significance of experimental data is usually based 
on a comparison of what actually happened to what would happen if chance 
alone were operating. Before applying statistics to the analysis of experi- 
mental da;a, the specific concept of probability should be understood as it 
applies to particular experimental data. Appendix III explains the concept of 
probability. The way in which an experiment is conducted usually determines 
not only whether inferences can be made but also the calculations required 
to make them. 



CONSUMER TESTING 

The consumer test is used to measure consumer acceptance of a prod- 
uct. Although the fate of a food product depends on consumer acceptance, 
formal studies of consumer preference are recent. Consumer studies are 
completely separate from laboratory panels, which do not attempt to predict 
consumer reaction. In this publication methods used in consumer tests are 
not described. Ideally a consumer test should cover a large sample of the 
population for whom the product is intended and should involve geographic 
as well as income sampling. Because of these conditions, consumer tests 
are often conducted by a specialized company. 



14 



APPENDIX I 

SAMPLE QUESTIONNAIRES AND EXAMPLES OF ANALYSES 

TRIANGLE TEST 
DIFFERENCE ANALYSIS 



DATE TASTER 

PRODUC T 



Instructions: Here are three samples for evaluation. Two of these samples 
are duplicates. Separate the odd sample for difference only. 

(1) (2) 

Sample Check odd sample 

314 



628 



542 



(3) Indicate the degree of difference between the duplicate samples and the 
odd sample. 

Slight Much 

Moderate Extreme 



(4) Acceptability: 

Odd sample more acceptable 



Duplicate samples more acceptable. 
(5) Comments: 



15 



Example: 

To determine if a difference existed between fish-potato flakes proc- 
essed under two different sets of conditions, a triangle test was used. 

The samples were first reconstituted by adding boiling water, then 
they were coded. On each tray there were three coded samples: two were 
the same and one was different. Eleven panelists were given two trays 
each, one after the other, and asked to identify the odd sample on each 
tray. This made a total of 22 judgments. 

The odd sample was correctly identified 19 times. According to Statis- 
tical Chart 1 of Appendix II, page 39, 19 correct judgments from 22 panel- 
ists in a triangle test are significant at the 0.1 percent level. The 
conclusion was that a difference existed between the samples. If the 
number of correct judgments had been less than 12, the conclusion would 
be that no detectable difference existed between the samples. 

The degree of difference indicated by those panelists who correctly 
chose the odd sample was: 

Slight = 1 Much = 6 

Moderate = 7 Extreme = 5 

The next part of the triangle test was to choose the more acceptable 
sample. Of the 19 panelists who correctly identified the odd sample, 14 
found the same sample more acceptable. According to Statistical Chart 1, 
for a two-sample test (there were only two choices at this point), this is 
below the number required for significance at the 5 percent level. However, 
the same degree of significance cannot be attached to the results of this 
secondary question as to the primary question. Once it is determined that 
a difference exists, another test should be conducted to determine which 
sample is more acceptable (probably a paired comparison test). 



16 



DUO-TRIO TEST 
DIFFERENCE ANALYSIS 



NAME 
DATE 



PRODUCT 



On your tray you have a marked control sample (R) and two coded 
samples, one is identical with R, the other is different. Which of the coded 
samples is different from R? 

SAMPLES CHECK ODD SAMPLE 

432 

701 



Example: 

To determine if methional could be detected when added to Cheddar 
cheese in amounts of 0.125 ppm and 0.250 ppm a duo-trio test was used. 
Each tray had a control sample marked R and two coded samples, one 
with methional added and one control. The duo-trio test was used in prefer- 
ence to the triangle test because less tasting is required to form a judgment 
using the duo-trio test. This fact becomes important when tasting a sub- 
stance with a lingering aftertaste, such as methional. 

The test was performed on two successive days using eight panelists. 
Each day the panelists were presented with two trays: one with 0.125 ppm 
and the other with 0.250 ppm methional added to a coded sample. This 
made a total of 16 judgments at each level. The results are shown in 
Table 1. 



17 











TABLE 1 




















Leve 


I of me 


thional 


added 


> PP 1 


Tl 




Panelists 




li 


st day 








2nd 


day 








0.125 




0.250 




0.125 




0.250 


PI 






X 




R 




R 






R 


P2 






R 




R 




R 






R 


P3 






X 




R 




X 






R 


P4 






R 




X 




X 






R 


P5 






R 




R 




R 






R 


P6 






X 




R 




X 






X 


P7 






R 




R 




R 






R 


P8 






R 




R 




R 






R 


Total 






5 




7 




5 






7 


P = Panelist 






















X = Wrong 






















R = Right 






















0.125 ppm = 10 


out 


of 16 


correct 
















0.250 ppm = 14 


out 


of 16 


correct 

















Consult Statistical Chart 1 of Appendix II, page 39, for 16 panelists 
in a two-sample test, which shows that 14 correct judgments are significant 
at the 1 percent level, while 10 are not significant, even at the 5 percent 
level. 

The conclusion is that methional added to Cheddar cheese can be 
detected at the 0.250 ppm level but not at the 0.125 ppm level. 



18 



NAME 



MULTIPLE COMPARISON 
DIFFERENCE ANALYSIS 



DATE 



QUESTIONNAIRE: 

You are receiving samples of . to compare for . 

You have been given a reference sample, marked R, to which you are to 
compare each sample. Test each sample; show whether it is better than, 
comparable to, or inferior to the reference. Then mark the amount of dif- 
ference that exists. 



Sample Number 
Better than R 
Equal to R 
Inferior to R 



AMOUNT OF DIFFERENCE 

None 

Slight 

Moderate 

Much 

Extreme 



COMMENTS: 

Any comments you may have about the flavor of the samples may be 
made here: 



Example: 

A multiple comparison test was conducted to determine how much anti- 
oxidant could be added to fish-potato flakes without tasters detecting a 



19 



difference in flavor. The flakes tested contained no antioxidant (0), 1 
unit, 2 units, 4 units, and 6 units of antioxidant. Rach tray contained a 
reference sample labeled R that contained no antioxidant and five coded 
samples (the four different levels of antioxidant and one sample with no 
antioxidant). Fifteen panelists were asked to evaluate these samples 
according to the score sheets on page 19. The ratings were given numerical 
values 1 to 9 by the person analyzing the results with "no difference" 
equaling 5, "extremely better than R" equaling 1, and "extremely inferior 
to R" equaling 9. The analysis of variance was calculated as shown in 
Table 2 and on the following pages. 

TABLE 2 



r^^n f*li etc 




Level of 


antioxidant 


added 




Total 


1 dll ClloLa 





1 Unit 


2 Units 


4 Units 


6 Units 


PI 


1 


4 


5 


1 


9 


20 


P2 


3 


3 


5 


5 


7 


23 


P3 


7 


3 


4 


4 


7 


25 


P4 


5 


7 


7 


3 


9 


31 


P5 


3 


3 


3 


3 


1 


13 


P6 


1 


1 


1 


1 


2 


6 


P7 


5 


5 


3 


5 


6 


24 


P8 


2 


2 


3 


2 


5 


14 


P9 


1 


3 


3 


3 


3 


13 


P10 


1 


1 


1 


7 


5 


15 


Pll 


6 


5 


1 


4 


1 


17 


P12 


7 


2 


1 


3 


9 


22 


P13 


3 


2 


3 


2 


6 


16 


P14 


3 


3 


1 


5 


1 


13 


P15 


3 


1 


5 


3 


3 


15 


Total 


51 


45 


46 


51 


74 


267 



P = Panelist 

Analysis of Variance 

Correction factor = (Total) 2 /Number of responses (15 tasters x 5 samples) 
CF = (267)V75= 71289/75= 950.52 

Sum of squares, samples = (Sum of the square of the total for each sample/ 

Number of judgments for each sample) — CF 

= [(512 + 452 + 462 + 512 + 742)/15] - CF 

= (14819/15)- CF = 987.93 - 950.52 

= 37.41 



20 



set up as follows: 


ss 


MS 


37.41 


9.35 


111.28 


7.95 


201.81 


3.60 


350.48 





Sum of squares, panelists = (Sum of the square of the total for each panel- 
ist/Number of judgments by each panelist) — CF 
= [(202 + 232 + 252 ... + 152)/5] - CF 

= (5309/5)- CF= 1061.80- 950.52 
= 111.28 

Total sum of squares - Sum of the square of each judgment — CF 

= (12 + 32 + 72 + . . . + 32)_ CF 
= 1301.00- 950.52 
= 350.48 

The analysis of variance chart was then 

Source of variance df SS MS F 

Samples 4 37.41 9.35 2.59* 

Panelists 14 111.28 7.95 2.21* 

Error 56 

Total 74 

df — Degrees of freedom for samples is the number of samples minus 
one. There were five samples, so the degrees of freedom for samples 
in this example is 4. Degrees of freedom for panelists is the number 
of panelists minus one. There were 15 panelists so the df for this 
source is 14. The df for total is the total number of judgments 
minus 1 (75- 1 = 74). 

Error— (l)To determine the df for "error" subtract the values obtained 
for the other variables (in this case 4 for samples and 14 for 
panelists) from the total, 74, 
i.e. 74- (4 f 14)= 56. 

(2) To determine the SS for "error" subtract the values obtained 
for the other variables (in this case 37.41 for samples and 
111.28 for panelists) from the total, 350.48, 
i.e. 350.48- (37.41 + 111.28)= 201.81. 

MS — The mean square for any variable is determined by dividing the SS 
by its respective degree of freedom. 

F — The variance ratio or F value for samples is determined by dividing 
the MS for samples by the MS for error or 9.35/3.60 = 2.59. The F 
value for panelists may be determined by dividing the MS for panel- 
ists by the MS for error. 



21 



To determine if the difference between the samples is significant, 

the calculated F value (2.59) is checked in Chart 2 of Appendix II on 

pages 40 and 41. With 4 degrees of freedom in the numerator and 56 degrees 

of freedom in the denominator, the variance ratio (F value) must exceed 

2.52 to be significant at the 5 percent level and it must exceed 3.65 to be 

significant at the 1 percent level. The value of 2.59 is therefore significant 

at the 5 percent level (*). If the variance ratio is not significant, the 

conclusion is that the addition of up to 6 units of antioxidant does not 

make a detectable difference in the flavor of antioxidant. 

Since there is a significant difference between the samples, the ones 
that are" different can be determined by using Duncan's Multiple Range 
Test (7, 16). 

1 Unit 2 Units 4 Units 6 Units 

Sample score = 51 45 46 51 74 

Sample mean - Score/Number of 

panelists = 51/15 45/15 46/15 51/15 74/15 
= 3.4 3.0 3.1 3.4 4.9 

The sample means are arranged according to magnitude: 

A B C D E 

6 Units 4 Units 2 Units 1 Unit 

4 - 9 3.4 3.4 3.1 3.0 

The standard error of the sample mean: 

SE = y/ (MS error/Number of judgments for each sample) 
= V (3.60/15) 
= V 0.24 = 0.49 

The "shortest significant ranges" for 2, 3, 4, and 5 means are deter- 
mined by using Chart 3 of Appendix II, on page 42, for the 5 percent level 
of probability and Chart 4 for the 1 percent level. In this case to determine 
the 5 percent level, Chart 3 is used to find the "Studentized ranges" rp for 
p - 2, . . . 5 means with 56 degrees of freedom (since 56 is not shown, the 
figure 60 is used). 

These values are then multiplied by the standard error of the mean, to 
obtain the shortest significant ranges, Rp. This gives: 

P 2 3 4 5 

rp (5 percent) 2.83 2.98 3.08 3.14 

Rp 1.39 1.46 1.51 1.54 



22 



The differences between the sample means are compared with the shortest 
significant range appropriate for the range under consideration in the fol- 
lowing order: 

(i) highest minus the lowest, highest minus second lowest, and so 
on to highest minus second highest; 

(ii) second highest minus lowest and so on to second highest minus 
third highest; 

(iii) and so on down to second lowest minus lowest. 

If, at any stage, a difference in (i) does not exceed the shortest sig- 
nificant range, the procedure stops, Say, for example, the highest mean 
minus the kth mean does not exceed the shortest significant range, then a 
line is drawn underscoring all means between the highest and the kth means 
inclusive, which indicates that this set of means should be grouped together 
as exhibiting no significant differences. 



Determination of shortest significant ranges 
Sample Sample Sample 



1 



Sample 
4 



Sample 
5 



Range a = 2 
b= 3 
c= 4 
d = 5 

(i) 

A-E= 4.9- 3.0= 1.9> 1.54 (R 5 ) 

A- D= 4.9- 3.1= 1.8> 1.51 (R 4 ) 

A- C = 4.9- 3.4= 1.5 > 1.46 (R 3 ) 

A- B= 4.9- 3.4= 1.5 > 1.39 (R 2 ) 

A is underscored as it is significantly different from the others. 



A BC DE 



(ii) 



B- E= 3.4- 3.0= 0.4 < 1.51 (1 4 ) 

Samples B, C, D, and E are underscored together as they exhibit no 
significant differences. 



23 



Proceed no further, since there is no significant difference between 
B and E. 

Therefore, the conclusion is that at the 5 percent level A is signif- 
icantly different from E or that 6 units made a significant difference 
in flavor from 0. This procedure may be repeated using Chart 4 if the 
samples were significantly different at the 1 percent level. 

This procedure can be repeated to see which panelists are signifi- 
cantly different from each other. 



RANKING 
DIFFERENCE ANALYSIS 



NAME 
DATE 



PRODUCT 



Evaluate these samples for tenderness. Please rank the samples for 
tenderness. The sample that is tenderest is ranked first, the second tender- 
est is ranked second, the toughest sample is ranked third. Place the code 
number in the appropriate box. 



Example: 

A ranking test was used to compare the texture of meat of three breeds 
of geese. The cooked meat was cut into pieces % inch by % inch by 1 
inch and one coded sample from each breed was presented to each of eight 
panelists. These panelists ranked the samples according to the score 
sheet above. 



24 



Results: 










Ranks: 




B, 


B 2 


B 3 




PI 


2 


1 


3 




P2 


2 


1 


3 




P3 


2 


1 


3 




P4 


1 


2 


3 




P5 


1 


3 


2 




P6 


2 


1 


3 




P7 


2 


1 


3 




P8 


1 


2 


3 




Total 


13 


12 


23 


P = Panelist 










B = Breed 










1 = First 










2 = Second 










3 = Third 











To analyze these results the ranks were transformed into scores, 
according to Fisher and Yates (9). Chart 5 of Appendix II, page 44, was 
used to determine the numerical value for each score. The sample ranked 
first of three samples was given a value of 0.85. When converting ranks, 
the middle rank is given a value of zero and the ranks beyond the middle 
are given negative values corresponding to the positive values given in 
the chart. In this case second is and third is —0.85. If we had six ranks 
the values would be: 

first = 1.27 

second = 0.64 

third = 0.20 

fourth = -0.20 

fifth = -0.64 

sixth =-1.27 

In Chart 5 values can be assigned to ranks in tests with up to 30 samples 
in the manner described above. 

Total 












cores 




B H 


B P 


B c 




PI 





0.85 


-0.85 




P2 





0.85 


-0.85 




P3 





0.85 


-0.85 




P4 


0.85 





-0.85 




P5 


0.85 


-0.85 







P6 





0.85 


-0.85 




P7 





0.85 


-0.85 




P8 


0.85 





-0.85 



Total 2.55 3.40 -5.95 



25 



The scores were then analyzed by the analysis of variance, as on page 20. 

CF = 
SS samples = ([2.552 + 3.402 + (-5.95)2] /8) _. CF 

= (53.465/8)- 

= 6.68 
SS panelists = 0/3 = 
Total SS = [02 + 02 + 02 + 0.852 . . . + (-0.85) 2 ] - CF 

= 11.56 



Variables 


df 


SS 




MS 


F 


Samples 


2 


6.68 


3.34 


9.54** 


Panelists 


7 











Error 


14 


4.88 


0.35 




Total 


23 


11.56 






Samples 


B H 






B p 


B c 




2.55 






3.40 


-5.95 


Mean samples 


0.32 






0.43 


-0.74 


A 


B 




C 






+ 0.43 


0.32 




-0.74 




Standard error 


/ (0.35/8) 


/0.04375 






0.209 












2 




3 






rp (5 percent) 


3.03 




3.18 






R P 


0.63 




0.66 







A - C = 0.43 - (-0.74) = 1.17 > 0.66 (R 3 ) 
A- B= 0.43 -0.32 = 0.11 < 0.63 (R 2 ) 

A _B C 

B-C=0.32 - (-0.74)= 1.06 > 0.63 (R 2 ) 

C is significantly different from A and B. 

The conclusion is that the meat from breed C was significantly less tender 
than that of breeds H and P at the 5 percent level. 

26 



SCORING 
DIFFERENCE TEST 



NAME 
DATE 



PRODUCT 



Evaluate these samples for flavor. Taste test each one. Use the ap- 
propriate scale to show your evaluation and check the point that best 
describes your feeling about the flavor of the sample. 



Code 
815 

Excellent 

.Very good 

. Good 

. Fair 

Poor 

Very poor 



Code 


558 


Excellent 


Very good 


Good 


Fair 


Poor 


Very poor 



Code 


394 


Excellent 


Very good 


Good 


Fair 


Poor 


Very poor 



Rea 



son 



Rea 



son 



Reas 



on 



27 



Example: 

Taste tests were conducted to determine any difference in flavor in 
the meat of three breeds of geese. The scoring test was used rather than 
multiple comparison as there was no standard or reference with which to 
compare. Eight panelists rated the coded samples according to the score 
sheet on page 27. The ratings were given numerical values by the person 
analyzing results with excellent = 1, very poor = 6. 

The results were analyzed by the analysis of variance, see page 20. 

Bj B 2 B 3 Total 

PI 3 2 3 8 

P2 4 6 4 14 

P3 3 2 3 8 

P4 1 4 2 7 

P5 2 4 2 8 

P6 1 3 3 7 

P7 2 6 4 12 

P8 Jl A _?_ 12. 

Total 18 33 23 74 

Correction factor = 742/24= 5476/24= 228.17 
SS samples = (1/8) (182 + 332 + 232) _ CF 

= (1/8) x 1942- CF 

= 242.75- 228.17 

= 14.58 
SS panelists = (1/3) (82 + I42 + . . . + 102) - CF 

= (1/3) x 730- CF 

= 243.33- 228.17 

= 15.16 
Total SS = (32 +42 + ... +22) -CF 

= 276- CF 

= 276- 228.] 7 

= 47.83 



Variables 


df 


SS 


MS 


F 


Samples 


2 


14.58 


7.29 


5.65* 


Panelists 


7 


15.16 


2.17 




Error 


14 


18.09 


1.29 




Total 


23 


47.83 







28 



There is a significant difference between samples at the 5 percent 
level. 

The Multiple Range Test is used to determine which samples are 
significantly different from the others. 





B, 


B 2 




B 3 


Mean samples 


= 18/8 


33/8 




23/8 




= 2.25 


4.13 




2.88 


Ranked means 


= A 


B 




C 




B 2 


B 3 




B! 




4.13 


2.88 




2.25 


Standard error 


= V (1.29/8) 
= 0.4 


= V 


0.16 





rp (5 percent) 3.03 3.18 

Rp 1.21 1.27 

A - C = 4.13 - 2.25 = 1.88 > 1.27 (R 3 ) 

A- B = 4.13- 2.88= 1.25 > 1.21 (R 2 ) 

A is significantly different from B and C. 

B- C = 2.88- 2.25 = 0.63 < 1.21 (R 2 ) 

A B C 

B and C are not significantly different from each other. The conclusion 
is that breed 2 is significantly different from breeds 1 and 3 at the 5 per- 
cent level. 



29 



SCORING 
DIFFERENCE TEST 



NAME 
DATE 



Evaluate these samples of goose meat for tenderness. Taste test 
each one. Use the appropriate scale to show your evaluation by checking 
at the point that best describes your feeling about the sample. 



Code 

664 

Extremely tender 

Very tender 

Moderately tender 

Slightly tender 

Slightly tough 

Moderately tough 

Very tough 

Extremely tough 



Code 

758 

Extremely tender 

Very tender 

Moderately tender 

Slightly tender 

Slightly tough 

Moderately tough 

Very tough 

Extremely tough 



Code 
708 
Extremely tender 

Very tender 

Moderately tender 

Slightly tender 

Slightly tough 

Moderately tough 

Very tough 

Extremely tough 



Reason 



Reason 



R 



eason 



Note: This is another example of scoring for difference. Analysis of vari- 
ance is used to analyze results (see page 20). 

Numerical values: extremely tender = 1 
extremely tough = 8 



30 



PAIRED COMPARISON 
DIFFERENCE TEST 



DATE TASTER 

PRODUCT 

Evaluate these two samples of peaches for texture. 

1. Is there a difference in texture between the two samples? 
Yes No 

2. Indicate the degree of difference in texture between the two samples by 
checking one of the following statements. 

846 is extremely better than 165 
846 is much better than 165 
846 is slightly better than 165 

No difference 

165 is slightly better than 846 
165 is much better than 846 
165 is extremely better than 846 

3. Rate the texture of the samples. 

165 846 

Good Good 

Fair Fair 



oor roor 



Poi 



Comments: 



31 



One of the tests conducted as part of a study of the effect of small 
doses of irradiation on the keeping quality of fresh peaches was a paired 
comparison test on texture. 

Four samples were compared: 
(1) control or krads sample, (2) 150 krads sample, (3) 200 krads sample, 
and (4) 250 krads sample. 

Each sample was compared with every other sample, making a total of six 
pairs. Each pair was presented to eight panelists for evaluation according 
to the score sheet on page 31. The experimental design requires that half 
the panelists taste one sample of the pair first and that the others taste 
the second sample first. 

The ratings of the panelists were given numerical values of +3, +2, +1, 
0,-1,-2,-3. Example: 

Sample Code 

150 krads 846 
1 P air= 250 krads 165 

The score sheet for the four panelists tasting sample 846 first was set up 
as follows with the values on the right being assigned by the analyzer. 

846 is extremely better than 165 (+3) 

846 is much better than 165 (+2) 

846 is slightly better than 165 ( + 1) 

No difference ( 0) 

165 is slightly better than 846 (-1) 

165 is much better than 846 (-2) 

165 is extremely better than 846 (-3) 

The four panelists who tasted sample 165 first received a score sheet set 
up as follows with the corresponding values on the right. 

165 is extremely better than 846 (+3) 

165 is much better than 846 (+2) 

165 is slightly better than 846 (+1) 

No difference. ( 0) 

846 is slightly better than 165 (-1) 

846 is much better than 165 (-2) 

846 is extremely better than 165 (-3) 

If one of the four panelists tasting 165 first indicated that 846 was much 
better than 165, his score was -2. 

The results of the scoring for all six pairs by all tasters was tabulated 
as shown in Table 3. 



32 















TABLE 3 










Order of 
presentation 


Fre 

-3 


quency i 
-2 -] 


j{ scores 
+1 


equal 
+2 


to 
+3 


Total 
score 


Mean 


Average 
preference 


0,150 
150,0 






3 


2 
1 




1 


1 




1 

-7 


0.25 
-1.75 


1.00 


0,200 
200,0 






2 




1 


1 
1 


2 
1 


« 


5 
-1 


1.25 
-0.25 


0.75 


0,250 
250,0 






2 






1 

2 


2 


1 


8 

-2 


2.00 
-0.50 


1.25 


150,200 
200,150 






1 


1 
1 


1 


2 
2 






-1 
1 


-0.25 
0.25 


-0.25 


150,250 
250,150 








3 


2 


1 
1 


1 




3 

-2 


0.75 
-0.50 


0.625 


200,250 
250,200 






1 


1 


2 
2 


1 


1 




-3 
3 


-0.75 
0.75 


-0.75 


Total 







9 


9 


8 


13 


8 


1 









Mean = Total score/Number of panelists = 1/4 = 0.25 

Average preferences = Vi (mean for 0,150 - mean for 150,0) 

= Vi (0.25-(-1.75)) = % (2.00) = 1.00 

The average preference of over 150 was 1.00 and the average preference 
of 150 over was -1.00. Thus the average preference of over 200 equals 
- the average preference of 200 over 0. 

The above data were used to perform an analysis of variance, accord- 
ing to the method of Scheffe (20). 

Main effects of treatments (&) were calculated by totaling the average 
preference of each sample over every other sample and dividing by the 
number of treatments. 

&o = M (average preference of over 150 + average preference of over 
200 + average preference of over 250) 

= % (1.00 +0.75 + 1.25) 

= % (3.00) =0.75 



33 



a 150 = Va (-1.00 - 0.25 + 0.625) = -0.15625 

a 2Q0 = M (-0.75 + 0.25 - 0.75) = -0.3125 

a 250 = Va (-1.25 - 0.625 + 0.75) = -0.28125 

The order effect (6) was calculated by totaling the mean for each order 
of each pair and dividing this total by the number of ordered pairs (6 pairs 
x 2 orders = 12). 

6 = (1/12) (0.25-1.75 + 1.25 -0.25 + 2.00 -0.50 -0.25 + 0.25 +0.75 -0.50 
-0.75 +0.75) 



= 0.104 



ANALYSIS OF VARIANCE TABLE 



Variables df SS MS F 



Main effects 


3 


24.4375 


8.1458 


Order effect 


1 


0.5192 


0.5192 


Error 


44 


74.0433 


1.6828 



4.84 



** 



Total 48 99.0000 



Sum of squares for main effects = number of panelists x number of treatments 
x sum of squares of each a 

= 8 x 4 x [0.75 2 + (-0.15625) 2 + (-0.3125) 2 + (-0.28125)?] = 24.4375 

Sum of squares for order effect = number of panelists x number of pairs 
x order effect {8 ) 

= 8 x 6 x 0.104 2 =0.5192 

Total sum of squares (using the frequency of each score as shown in Table 3) 
= 3 2 (0 + 1) + 2 2 (9 + 8) + 1 2 (9 + 13) + 2 (8) 
= 99.00 

Sum of squares for error = 99.00 - 24.4375 - 0.5192 = 74.0433 

di-Main effects. There were 4 treatments, so the degrees of freedom = 3. 

Order effect. There were 2 orders — one sample of pair first or the other 
sample of the pair first, df = 1. 

Total. For paired comparison, the df for total is the total number of 
observations = 48. 

Error. 48 - 3 - 1 = 44. 

MS for main effects = 24.4375/3 = 8.1458 
MS for order effect = 0.5192/1 = 0.5192 
MS for error = 74.0433/44 = 1.6828 



34 



The F ratio is determined for each variable by dividing its MS by the 
MS error. Consult Chart 2 on pages 40 and 41 to see if the F ratio is signi- 
ficant. This procedure is described on page 22. In our example, the main 
effects are significantly different at the 1 percent level. 

Use Tukey's Test to determine which samples are different. Because 
Duncan's Multiple Range Test (7) has previously been used and explained 
in this booklet (see pages 22-23), it is used here to determine which sam- 
ples are different. 

150 200 250 



Average preferences 



0.75 



a i50 

-0.15625 



a 200 

-0.3125 



a 250 

-0.28125 



B 



D 



a o 
0.75 



a 



150 



-0.15625 



a 



250 



-0.28125 



a 



200 



-0.3125 



S.E . =\/ (MS error/Number of judgments for each sample) = \/ (1.6828/24) 
= y/ 0.070 =0.27 



rp (5 percent) 
R P 



2 

2.86 
0.77 



3 

3.01 
0.81 



4 

3.10 
0.84 



61 o ~ Soo = °- 75 ~ (-0-3125) = 1.0625 > 0.84 






250 



= 0.75 - (-0.28125) = 1.03125 > 0.81 



a - a 150 = 0.75 - (-0.15625) = 0.90625 > 0.77 

S iso " a 200 = "0-15625 - (-0.3125) = 0.15626 < 0.81 



a. 



a 



150 



a 



250 



a 



200 



The control sample is significantly different from the other samples. Its 
score is higher and therefore it would be considered to have better texture. 
This does not mean necessarily that the other samples are of poor texture. 
To determine the quality of the samples, the ratings given them by the 
panelists were tabulated to see if they were considered to be good, fair, or 
poor. An average score can be determined for each sample by assigning the 
values good = 3, fair = 2, poor = 1, and computing the average for each 
sample. 



35 



HEDONIC SCALE 
SCORING 



DATE 



TASTER 



PRODUCT 



Taste test these samples and check how much you like or dislike each 
one. Use the appropriate scale to show your attitude by checking at the 
point that best describes your feeling about the sample. Please give a 
reason for this attitude. Remember you are the only one who can tell what 
you like. An honest expression of your personal feeling will help us. 



CODE 
459 



CODE 
667 



CODE 

619 



CODE 
347 



Like 
extremely 

Like 
very much 

Like 
moderately 

Like 
slightly 

Neither like 
nor dislike 

Dislike 
slightly 

Dislike 
moderately 

Dislike 
very much 

Dislike 
extremely 

REASON 



Like 
extremely 

Like 
very much 

Like 
moderately 

Like 
slightly 

Neither like 
nor dislike 

Dislike 
slightly 

Dislike 
moderately 

Dislike 
very much 

Dislike 
extremely 

REASON 



.Like 

extremely 

Like 
very much 

Like 
moderately 

.Like 

slightly 

Neither like 
nor dislike 

Dislike 
slightly 

. Dislike 
moderately 

_Dislike 
very much 

Dislike 
ext remely 

REASON 



Like 
extremely 

Like 
very much 

Like 
moderately 

Like 

slightly 

Neither like 
nor dislike 

Dislike 
slightly 

.Dislike 
moderately 

Dislike 
very much 

Dislike 

extremely 

REASON 



Note: To analyze the results of this test use analysis of variance (see 
pages 20-23). 

Numerical values: like extremely = 9 
dislike extremel) = 1 



36 



PAIRED COMPARISON 
PREFERENCE 

DATE TASTER 

PRODUCT 



INSTRUCTIONS: (A) Here are two samples for evaluation. Please indicate 
which sample you prefer. 

622 244 

(B) Indicate the degree of preference between the two 
samples. 



Slight 



Moderate 
Much 



Extreme 



Example: 

To determine which of two samples of Cheddar cheese (Number 1 or 
Number 2) had more acceptable flavor, a paired comparison test was used. 
The samples were coded and one sample of each cheese was presented on 
each tray. Eight panelists were presented with three trays each, one after 
the other. This made a total of 24 judgments. 

Cheese Number 2 was preferred 17 times in the 24 judgments. Accord- 
ing to Statistical Chart 1 of Appendix II, on page 39, in the two-sample 
test, one sample must be preferred 18 times to be significantly more 
acceptable. So we conclude that neither cheese was significantly more 
acceptable for flavor. 



37 



RANKING 
PREFERENCE 



NAME 
DATE 



PRODUCT. 



Please rank these samples according to your preference. 

Code 



First 

Second 

Third 



Fourth 

Note: The results are analyzed by converting ranks to scores and con- 
ducting analysis of variance as shown for Ranking Difference Analysis, 
page 24. 



38 



APPENDIX II 
STATISTICAL CHART 1 



Number 

of 
tasters 


Two-sample test 
number of concurring c 
necessary to establ 
significance 


rhoices 
lish 


Triangle test 

difference analysis, 

number of correct answers 

necessary to establish 

significance 




* 


** 


*** 


* 


** 


*** 


1 
2 
3 
4 
5 


— 


— 


— 


3 

4 
5 


— 


- 


- 


- 


- 


5 


- 


6 


6 


— 


— 


5 


6 


— 


7 


7 


— 


— 


5 


6 


7 


8 


8 


8 


— 


6 


7 


8 


9 


8 


9 


— 


6 


i 


8 


10 


9 


10 


— 


7 


8 


9 


11 


10 


11 


11 


7 


8 


10 


12 


10 


11 


12 


8 


9 


10 


13 


11 


12 


13 


8 


9 


11 


14 


12 


13 


14 


9 


10 


11 


15 


12 


13 


14 


9 


10 


12 


16 


13 


14 


15 


9 


11 


12 


17 


13 


15 


16 


10 


11 


13 


18 


14 


15 


17 


10 


12 


13 


19 


15 


16 


17 


11 


13 


14 


20 


15 


17 


18 


11 


13 


14 


21 


16 


17 


19 


12 


13 


15 


22 


17 


18 


19 


12 


14 


15 


23 


17 


19 


20 


12 


14 


16 


24 


18 


19 


21 


13 


15 


16 


25 


18 


20 


21 


13 


15 


17 


26 


19 


20 


22 


14 


15 


17 


27 


20 


21 


23 


14 


16 


18 


28 


20 


22 


23 


15 


16 


18 


29 


21 


22 


24 


15 


17 


19 


30 


21 


23 


25 


15 


17 


19 


31 


22 


24 


25 


16 


18 


20 


32 


23 


24 


27 


16 


18 


20 


33 


23 


25 


27 


17 


18 


21 


34 


24 


25 


27 


17 


19 


21 


35 


24 


26 


28 


17 


19 


22 


36 


25 


27 


29 


18 


20 


22 


37 


25 


27 


29 


18 


20 


22 


3 8 


26 


28 


30 


19 


21 


23 


39 


27 


28 


31 


19 


21 


23 


40 


27 


29 


31 


19 


21 


24 


41 


27 


29 


32 


20 


22 


24 


42 


28 


30 


32 


20 


22 


25 


43 


28 


30 


33 


21 


23 


25 


44 


29 


31 


33 


21 


23 


25 


45 


30 


32 


34 


22 


24 


26 


46 


. 30 


32 


35 


22 


24 


26 


47 


31 


33 


35 


23 


24 


27 


48 


31 


33 


36 


23 


25 


27 


49 


32 


34 


37 


23 


25 


28 


50 


32 


35 


37 


24 


26 


28 



5 percent level of significance. ** 1 percent level. ***0.1 percent level. 



39 



STATISTICAL CHART 2 

Variance Ratio — 5 Percent Points for Distribution of F 
n^ — Degrees of freedom for numerator 
n — Degrees of freedom for denominator 



n 2 \ 


1 


2 


3 


4 


5 


6 


8 


12 


24 


oo 


1 


161.4 


199.5 


215.7 


224.6 


230.2 


234.0 


238.9 


243.9 


249.0 


254.3 


2 


18.51 


19.00 


19.16 


19.25 


19.30 


19.33 


19.37 


19.41 


19.45 


19.50 


3 


10.13 


9.55 


9.28 


9.12 


9.01 


8.94 


8.84 


8.74 


8.64 


8.53 


4 


7.71 


6.94 


6.59 


6.39 


6.26 


6.16 


6.04 


5.91 


5.77 


5.63 


5 


6.61 


5.79 


5.41 


5.19 


5.05 


4.95 


4.82 


4.68 


4.53 


4.36 


6 


5.99 


5.14 


4.76 


4.53 


4.39 


4.28 


4.15 


4.00 


3.84 


3.67 


7 


5.59 


4.74 


4.35 


4.12 


3.97 


3.87 


3.73 


3.57 


3.41 


3.23 


8 


5.32 


4.46 


4.07 


3.84 


3.69 


3.58 


3.44 


3.28 


3.12 


2.93 


9 


5.12 


4.26 


3.86 


3.63 


3.48 


3.37 


3.23 


3.07 


2.90 


2.71 


10 


4.96 


4.10 


3.71 


3.48 


3.33 


3.22 


3.07 


2.91 


2.74 


2.54 


11 


4.84 


3.98 


3.59 


3.36 


3.20 


3.09 


2.95 


2.79 


2.61 


2.40 


12 


4.75 


3.88 


3.49 


3.26 


3.11 


3.00 


2.85 


2.69 


2.50 


2.3D 


13 


4.67 


3.80 


3.41 


3.18 


3.02 


2.92 


2.77 


2.60 


2.42 


2.21 


14 


4.60 


3.74 


3.34 


3.11 


2.96 


2.85 


2.70 


2.53 


2.35 


2.13 


15 


4.54 


3.68 


3.29 


3.06 


2.90 


2.79 


2.64 


2.48 


2.29 


2.07 


16 


4.49 


3.63 


3.24 


3.01 


2.85 


2.74 


2.59 


2.42 


2.24 


2.01 


17 


4.45 


3.59 


3.20 


2.96 


2.81 


2.70 


2.55 


2.38 


2.19 


1.96 


18 


4.41 


3.55 


3.16 


2.93 


2.77 


2.66 


2.51 


2.34 


2.15 


1.92 


19 


4.38 


3.52 


3.13 


2.90 


2.74 


2.63 


2.48 


2.31 


2.11 


1.88 


20 


4.35 


3.49 


3.10 


2.87 


2.71 


2.60 


2.45 


2.28 


2.08 


1.84 


21 


4.32 


3.47 


3.07 


2.84 


2.68 


2.57 


2.42 


2.25 


2.05 


1.81 


22 


4.30 


3.44 


3.05 


2.82 


2.66 


2.55 


2.40 


2.23 


2.03 


1.78 


23 


4.28 


3.42 


3.03 


2.80 


2.64 


2.53 


2.38 


2.20 


2.00 


1.76 


24 


4.26 


3.40 


3.01 


2.78 


2.62 


2.51 


2.36 


2.18 


1.98 


1.73 


25 


4.24 


3.38 


2.99 


2.76 


2.60 


2.49 


2.34 


2.16 


1.96 


1.71 


26 


4.22 


3.37 


2.98 


2.74 


2.59 


2.47 


2.32 


2.15 


1.95 


1.69 


27 


4.21 


3.35 


2.96 


2.73 


2.57 


2.46 


2.30 


2.13 


1.93 


1.67 


28 


4.20 


3.34 


2.95 


2.71 


2.56 


2.44 


2.29 


2.12 


1.91 


1.65 


29 


4.18 


3.33 


2.93 


2.70 


2.54 


2.43 


2.28 


2.10 


1.90 


1.64 


30 


4.17 


3.32 


2.92 


2.69 


2.53 


2.42 


2.27 


2.09 


1.89 


1.62 


40 


4.08 


3.23 


2.84 


2.61 


2.45 


2.34 


2.18 


2.00 


1.79 


1.51 


60 


4.00 


3.15 


2.76 


2.52 


2.37 


2.25 


2.10 


1.92 


1.70 


1.39 


120 


3.92 


3.07 


2.68 


2.45 


2.29 


2.17 


2.02 


1.83 


1.61 


1.25 


oo 


3.84 


2.99 


2.60 


2.37 


2.21 


2.09 


1.94 


1.75 


1.52 


1.00 



40 



STATISTICAL CHART 2 - Concluded 

Variance Ratio — 1 Percent Points for Distribution of F 
n^ — Degrees of freedom for numerator 
rc 2 — Degrees of freedom for denominator 



n 2 ^\ 


1 


2 


3 


4 


5 


6 


8 


12 


24 


so 


1 


4052 


4999 


5403 


5625 


5764 


5859 


5981 


6106 


6234 


6366 


2 


98.49 


99.00 


99.17 


99.25 


99.30 


99.33 


99.36 


99.42 


99.46 


99.50 


3 


34.12 


30.81 


29.46 


28.71 


28.24 


27.91 


27.49 


27.05 


26.60 


26.12 


4 


21.20 


18.00 


16.69 


15.98 


15.52 


15.21 


14.80 


14.37 


13.93 


13.46 


5 


16.26 


13.27 


12.06 


11.39 


10.97 


10.67 


10.29 


9.89 


9.47 


9.02 


6 


13.74 


10.92 


9.78 


9.15 


8.75 


8.47 


8.10 


7.72 


7.31 


6.88 


7 


12.25 


9.55 


8.45 


7.85 


7.46 


7.19 


6.84 


6.47 


6.07 


5.65 


8 


11.26 


8.65 


7.59 


7.01 


6.63 


6.37 


6.03 


5.67 


5.28 


4.86 


9 


10.56 


8.02 


6.99 


6.42 


6.06 


5.80 


5.47 


5.11 


4.73 


4.31 


10 


10.04 


7.56 


6.55 


5.99 


5.64 


5.39 


5.06 


4.71 


4.33 


3.91 


11 


9.65 


7.20 


6.22 


5.67 


5.32 


5.07 


4.74 


4.40 


4.02 


3.60 


12 


9.33 


6.93 


5.95 


5.41 


5.06 


4.82 


4.50 


4.16 


3.78 


3.36 


13 


9.07 


6.70 


5.74 


5.20 


4.86 


4.62 


4.30 


3.96 


3.59 


3.16 


14 


8.86 


6.51 


5.56 


5.03 


4.69 


4.46 


4.14 


3.80 


3.43 


3.00 


15 


8.68 


6.36 


5.42 


4.89 


4.56 


4.32 


4.00 


3.67 


3.29 


2.87 


16 


8.53 


6.23 


5.29 


4.77 


4.44 


4.20 


3.89 


3.55 


3.18 


2.75 


17 


8.40 


6.11 


5.18 


4.67 


4.34 


4.10 


3.79 


3.45 


3.08 


2.65 


18 


8.28 


6.01 


5.09 


4.58 


4.25 


4.01 


3.71 


3.37 


3.00 


2.57 


19 


8.18 


5.93 


5.01 


4.50 


4.17 


3.94 


3.63 


3.30 


2.92 


2.49 


20 


8.10 


5.85 


4.94 


4.43 


4.10 


3.87 


3.56 


3.23 


2.86 


2.42 


21 


8.02 


5.78 


4.87 


4.37 


4.04 


3.81 


3.51 


3.17 


2.80 


2.36 


22 


7.94 


5.72 


4.82 


4.31 


3.99 


3.76 


3.45 


3.12 


2.75 


2.31 


23 


7.88 


5.66 


4.76 


4.26 


3.94 


3.71 


3.41 


3.07 


2.70 


2.26 


24 


7.82 


5.61 


4.72 


4.22 


3.90 


3.67 


3.36 


3.03 


2.66 


2.21 


25 


7.77 


5.57 


4.68 


4.18 


3.86 


3.63 


3.32 


2.99 


2.62 


2.17 


26 


7.72 


5.53 


4.64 


4.14 


3.82 


3.59 


3.29 


2.96 


2.58 


2.13 


27 


7.68 


5.49 


4.60 


4.11 


3.78 


3.56 


3.26 


2.93 


2.55 


2.10 


28 


7.64 


5.45 


4.57 


4.07 


3.75 


3.53 


3.23 


2.90 


2.52 


2.06 


29 


7.60 


5.42 


4.54 


4.04 


3.73 


3.50 


3.20 


2.87 


2.49 


2.03 


30 


7.56 


5.39 


4.51 


4.02 


3.70 


3.47 


3.17 


2.84 


2.47 


2.01 


40 


7.31 


5.18 


4.31 


3.83 


3.51 


3.29 


2.99 


2.66 


2.29 


1.80 


60 


7.08 


4.98 


4.13 


3.65 


3.34 


3.12 


2.82 


2.50 


2.12 


1.60 


120 


6.85 


4.79 


3.95 


3.48 


3.17 


2.96 


2.66 


2.34 


1.95 


1.38 


oo 


6.64 


4.60 


3.78 


3.32 


3.02 


2.80 


2.51 


2.18 


1.79 


1.00 



41 



< 

X 

u 
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< 
y 

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3 5?3 
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r- 


t - 


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r - 


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o 


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ce 


CO 


CC 


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co 


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CO 


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r- 


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(N 


C^J 


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o 


ON 


ON 


CO 


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10 


IC 


CO 


r _, 


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o\ 


o 


o 


lO 


o 


CO 


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IO 


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-t 


— »■ 


-+ 


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-t 


•^ 


-4. 


-* 


CO 


ro 


CO 


CO 


CO 


CO 


CO 


co 


CO 


CO 


CM (M 




00 


NO 


-+ 


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CO 


CC 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


cc 


CC 


CO 


CO 


CO 


CO 


co 


CO 


CO 


CO 


CO 


CO CO 






On 


O 


Cl 


cc 


cc 


rH 


NO 


CI 


r~ 


IC 


"+ 


CI 


r— 


o 


O 


cc, 


r- 


r- 


vo 


1/3 


rf 


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m 


CI 


o 


CO 


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o 


o 


lO 


o 


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VO 


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in 


in 


<* 


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-c- 


-+ 


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CO 


CO 


co 


CO 


CO 


CO 


CO 


M 


CO 


CO 


CO 


Cl 


CM CJ 




CO 


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CO 


CO 


CO 


CO 


co 


CO 


co 


CC 


co 


co 


CO 


CO. 


CO 


CO 


CO 


CO 


CO 


CO. 


CO 


CO 


CO 


CO 


CO 


CO CO 






ON 


o 


CM 


CO 


CO 


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_ 


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00 


r^ 


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m 


m 


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CM 


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cc 


a 


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«* 


CM Ov 




o 


o 


lO 


O 


cc 


NO 


NO 


i.O 


LO 


** 


"* 


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-+ 


CO 


CO 


CO 


CO' 


co 


CO 


CO 


co 


CO 


CO 


co 


CI 


01 


Cl 


CM — 




CO 


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CO 


CC 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


co 


CO 


CO 


CO 


CO 


cc 


CO 


CO 


CO 


CO 


CO 


CO 


CO CO 






ON 


o 


Cl 


c^ 


CO 


re 


1' 


o 


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o 


cc 


h- 


vn 


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CO 


CI 


_ 


o 


o 


CO 


r- 


vn 


iC 


Cl 


r~. 


co m 


vo 


o 


o 


in 


o 


cc 


NO 


VO 


m 


LO 


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-+■ 


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co 


co 


CO 


CO 


CO 


CO 


CO 


CC 


CI 


CI 


CJ 


CI 


CJ 


CI 


Cl 


r-^ ^^ 


00 


NO 


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X* 


co 


CC 


CO 


CO 


CO 


co 


CO 


CO 


cc 


CO 


co 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO CO 






On 


o 


Cl 


a~- 


CO 


CO 


CI 


r - 


CO 


ON 


NO 


IC 


CO 


— 


o 


CO 


1 - 


vC 


IC 


«f 


CI 


p-i 


cc 


o 


r- 


-* 


CM On 




o 


o 


1/5 


O 


cc 


NO 


LO 


l.O 


«* 


«* 


co 


CO 


CO 


co 


CO 


CO 


CM 


CI 


CI 


C) 


CM 


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CI 


CM 


01 


1— 




— CO- 




00 


NO 


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co 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CC 


CO 


CO 


CO 


CO 


co 


CO 


CC 


CO 


CO 


c^ 


cc 


CO CO 






On 


O 


CI 


ON 


X* 


«# 


r- 


M 


r- 


m 


CO 


o 


r~ 


IC 


CO 


CM 


, 


o> 


CO 


r- 


IC 


^. 


CO 


CM 


o 


CO 


m cm 




o 


o 


lO 


O 


1 - 


NO 


lO 


-t 


"* 


CO 


CO 


CO 


CO 


CI 


<N 


CI 


CI 


CI 


















o 


o o 




CO 


vn 


-t 


•* 


CO 


CO 


CO 


co 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


co 


co 


CO 


CO 


CO 


CO 


CO 


CO 


CO 


co 


co co 






ON 


o 


p_i 


■=* 


CO 


1^- 


ON 


"+ 


o 


r- 


CO 


,_ 


00 


VO 


in 


c^ 


CM 


,_ 


a 


on 


r~- 


NO 


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■e. 


— 


cc 


in ci 




o 


o 


1/3 


o 


o 


1/5 


-+ 


CC 


CO 


CO 


CM 


CI 


a 


1— 1 


i — 


i — i 


i — 1 


1 — 1 


1 — 1 


— 1 


o 


o 


o 


o 


o 


o 


CC 


On On 




00 


NO 


— f 


-+ 


CO 


co 


CO- 


CO 


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CO 


CO 


CO 


CO 


CO 


CO 


CO 


CO' 


CO 


CO 


CO 


CO 


CO 


co 


CO 


CO 


CI 


CJ CM 






On 


O 


CO 


rf 


VO 


io 


VO 


o 


in 


_ 


CO 


NO 


CO 


,_, 


o 


CO 


r~ 


vn 


m 


CO 


CI 


r _ 


o 


cc 


vn 


M 


o tr- 




o 


O 


W 


o> 


vO 


«* 


CO 


CM 


CI 


i — i 


i-H 


o 


o 


o 


O 


CC' 


O 


O 


0> 


ON 


ON 


CC 


ON 


OC 


X 


cc 


X 


ee r- 




CO 


vO 


-+ 


CO 


co 


CC 


CO 


CO 


CO 


CO 


CC 


CO 


CO 


CO 


CO 


CO 


Cl 


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CM 


CM 


C) 


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CJ 


CM 


CM 


Cl 


Cl 


CM CM 


Q. / 




CM 


CO 


■<t 


IO 


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r~- 


30 


Ov 


s 




CJ 


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- 1 - 


m 


vO 


r-- 


X 


c 


— 


CM 




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~r. 


o 


— 


O 


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/ c 








































CI 


CM 


CM 


CI 


CI 


CO 


— 


-o 


o 8 



42 



< 

X 

u 
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< 
y 

< 



Du 



en m E— i 



H 


v: 




U. 


e 


CO 


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l> 








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^ 


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3 


c/5 


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NO 


lO 


r- 


o 


-+" 


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to 


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IO 


rf 


CO 


Ol 


,_, 


ON 


NO 


IO 


00 


o 


O 


o 


CO 


lO 


co 


CO 


o 


co 


r- 


m 


co 


Ol 


l-H 


o 


o 


o 


CO 


CO 


co 


t - 


I - 


f- 


1 - 


t^ 


r- 


NO 


NO 


NO 


NO 


o 


o 


rf 


On 


r- 


no 


VO 


NO 


m 


IO 


m 


10 


IO 


IO 


lO 


to 


-+ 


rt 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


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rf 


1-1 


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1-1 












































































in 


On 


NO 


IO 


r- 


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-+ 


ON 


to 


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to 


rf 


CO 


CM 


pH 


O 


NO 


rf 


O 


o 


o 


o 


co 


lO 


CO 


CO 


o 


co 


r- 


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CO 


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' — 1 


o 


o 


o 


cc 


CO 


co 


t - 


t - 


t - 


O- 


O. 


t- 


NO 


NO 


NO 


NO 


lO 


o 

On 


rf 


On 


f~ 


NO 


NO 


NO 


lO 


tO 


in 


LO 


LO 


tO 


lO 


m 


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rf 


rf 


rf 


rt 


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rf 


rf 


rf 


rf 


rf 


rf 


rf 






















tO 


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r- 


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Ol 


r> 


r- 


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On 


co 


2 


^ 


o 


O 


o 


CO 


1/5 


00 


CO 


o 


CO 


t^ 


to 


n 


<— 1 


O 


o 


o 


co 


co 


CO 


t- 


t - 


t- 


NO 


N0 


NO 


10 


lO 


rf 


cs 


O 


rt 


On 


r~ 


NO 


NO 


NO 


LO 


m 


LO 


lO 


lO 


to 


lO 


LO 


-+ 


-t 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 






















rf 


co 


rf 


rt 


NO 


On 


co 


on 


rf 


_ 


CO 


-+ 


r~> 


r^ 


IO 


CO 


r^ 


O 


m 


CO 


co 


o 


o 


CO 


lO 


r^ 


co 


o 


CO 


r- 


lO 


CO 


Ol 


1—1 


o 


O 


On 


co 


co 


cc 


t - 


t - 


t - 


NO 


O 


NO 


to 


tO 


rf 


CO 




o 

On 


rf 


On 


t-^ 


NO 


NO 


NO 


IO 


LO 


LO 


LO 


to 


to 


LO 


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-+ 


rf 


rf 


rf 


rf 


rf 


if 


rf 


rf 


rf 


rf 


rf 


rf 






















CO 


rt 


c\ 


CO 


-f 


r- 


^H 


NO 


Ol 


On 


NO 


,_, 


1^- 


IO 


CM 


,_| 


rf 


r- 


CM 


rf 


VO 


O 


o 


Ol 


rt 


r^ 


Ol 


o 


r~ 


NO 


•H. 


CO 


Ol 


1 — 1 


o 


o 


o 


CC 


CO 


f- 


t - 


i - 


NO 


NO 


NO 


NO 


to 


rf 


rf 


CO 


^ H 


d 

on 


rf 

1 — 1 


O 


t^ 


nO 


NO 


lO 


to 


m 


LO 


tO 


tO 


LO 


lO 


it 


-t 


-+ 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 






















Ol 


co 


r- 


CO 


o 


i+ 


co 


CO 


On 


^r> 


c^ 


co 


•+ 


OJ 


O 


00 


r— 


•* 


CO 


*—< 




o 


o 


1 ' 


rt 


no 


OJ 


On 


l> 


LO 


rt 


Ol 


1 — 


o 


o 


o 


cc 


CC 


r- 


t - 


I - 


NO 


NO 


NO 


NO' 


to 


lO 


rf 


CO 


CO 




o 

O 


rt 


On 


t> 


NO 


NO 


lO 


lO 


1/5 


10 


lO 


IO 


IO 


lO 


** 


-* 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 






















NO 


Tf 


co 


rt 


NO 


o 


-f 


O 


NO 


CM 


0!N 


to 


CM 


en 


NO 


rt 


NO 


ON 


IO 


NO 


CM 


O 


o 


o 


CO 


NO 


' — 1 


■co 


NO 


lO 


CO 


CM 


1 — i 


o 


O 


o 


co 


co 


t- 


r- 


NO 


VO 


NO 


LO 


lO 


LO 


rf 


CO 


CO 


CM 


1—1 


O 

o\ 


rf 


On 


t- 


NO 


NO 


lO 


lO 


LO 


in 


lO 


lO 


lO 


-* 


Tf 


-* 


■* 


rf 


rt 


rf 


rf 


rt 


rf 


rf 


rf 


rf 


rf 


rf 


rf 






















CO 


in 


r- 


00 


^ 


1* 


ON 


to 


_ 


r- 


1(0 


ro 


r^ 


co 


,_ 


m 


_ 


rf 


PA 


O 


O 


o 


o 


O 


co 


10 


o 


co 


lO 


-I- 


Ol 


1 — i 


O 


o 


O 


cc. 


1 - 


[ - 


r- 


NO 


NO 


NO 


to 


LO 


to 


rf 


rf 


CO 


OJ 


CM 




o 

ON 


rf 


On 


t~- 


NO 


NO 


m 


IO 


m 


lO 


to 


LO 


»* 


■* 


-* 


-* 


-* 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 












rt 


o 


co 


I— 1 


NO 


«* 


Ol 


Ol 


-t- 


1^ 


^H 


NO 


OJ 


co 


rf 


,_, 


r^ 


<y 


O 


r^ 


to 


h- 


i — i 


IO 


r- 




o 


o 


O 


Cl 


rt 


o 


1 - 


10 


CO 


CM 


'—< 


o 


O 


co 


CC 


I - 


r~ 


NO 


NO 


NO 


lO 


lO 


to 


rf 


rt 


co 


c^ 


O) 


i— i 




o 

ON 


rf 

1 — 1 


On 


t^ 


O 


NO 


in 


lO 


in 


LO 


lO 


to 


-* 


rj> 


It 


-t 


rt 


rf 


rf 


rf 


rf 


rt 


rf 


rf 


rf 


rf 


rf 


rf 


rt 












o 


IP 


o 


r- 


CM 


o 


NO 


NO 


CO 


CCN 


l-~ 


Ol 


on 


rf 


^ 


00 


co 


ON 


r? 


r<~ 


,_ 


rf 


r^ 


rt 


rf 




o 


o 


On 


Ol 


rt 


o 


NO 


rt 


CO 


Ol 


o 


o 


CO 


CO 


t - 


t - 


NO 


NO 


NO 


lO 


LO 


rf 


rf 


rf 


CO 


OJ 


OJ 






o 


rf 


co 


r^ 


vO 


lO 


in 


m 


in 


IO 


IO 


rf 


-f 


-f 


«* 


it 


-t 


-f 


rf 


rf 


-+ 


rf 


rf 


rf 


rf 


rf 


rf 


rf 


rf 




On 


rH 


































































CO 


co 


,-H 


o 


lO 


co 


_ 


CI 


-f 


00 


Ol 


i - 


CO 


O 


NO 


co 


CO 


rf 


,_, 


o> 


NO 


o 


c^ 


r^ 


On 




o 


o 


On 


1 — i 


CO 


CC 


NO 


rt 


Ol 


i — i 


o 


ON 


co 


t^ 


i- 


NO 


NO 


lO 


lO 


10 


rf 


rf 


rf 


co 


co 


CO 


CM 


i — i 


O 




o 

On 


rf 


co 


o 


NO 


lO 


LO 


m 


m 


m 


in 


rf 


-+ 


-t 


-+ 


Tt 


rf 


rt 


rf 


rf 


rf 


rt 


rf 


rf 


rf 


rf 


rf 


rf 


rf 












NO 


pH 


CO 


Ol 


t- 


NO 


rt 


rt 


-* 


o 


-+ 


O 


NO 


co 


O 


r- 


CM 


ON 


NO 


rf 


Ol 


-* 


r - 


rt 


rt 




O 


o 


CO 


1 — ' 


Ol 


co 


l/J 


CO 


1—1 


o 


o 


co 


i - 


t - 


NO 


NO 


lO 


in 


LO 


rf 


rf 


co 


CO 


CO 


ro 


Ol 




i — i 


o 




O 


rf 


CO 


r» 


NO 


LO 


lO 


lO 


m 


IO 


rt 


-+ 


-t- 


■*t 


If 


-f 


rt 


rt 


rf 


rf 


*+ 


rf 


rf 


rf 


rt 


rf 


rf 


rf 


rf 




O 


rH 


































































CC 


CO 


in 


CO 


co 


NO 


NO 


NO 


On 


n 


CT: 


-f 


o 


NO 


co 


§ 


NO 


c^ 


o 


00 


CM 


r^ 


CM 


NO 


no 




O 


o 


r^ 


o 


1 ' 


1 - 


rt 


Ol 


o 


O 


CO 


1 - 


NO 


NO 


IO 


lO 


to 


rf 


rf 


CO 


CO 


CO 


Ol 


CM 


l-H 


i — l 


o 


On 




O 
On 


rf 

1 — 1 


CC 


r- 


NO 


lO 


m 


lO 


m 


rt 


rt 


rt 


Tt 


rf 


It 


rt 


-t 


rf 


rf 


rt 


rf 


rf 


rf 


rt 


rf 


rf 


rf 


rt 


CO 












l-H 


m 


r- 


rf 


On 


co 


l - 


CO' 


CM 


to 


o 


m 


,— i 


CO 


m 


ec 


00 


rf 


i — < 


>cn 


NO 


o 


r-- 


CO 


O 


rf 


O 


c_> 


NO 


o> 


1 — 1 


NO 


CO 


OH 


On 


DO 


i - 


NO 


NO 


1/5 


IO 


rt 


*+ 


CO 


rt 


co 


Ol 


CM 


CM 








O 


o 


On 




O 


-f 


CO 


NO 


vO 


m 


lO 


in 


-f 


rt 


-+ 


rt 


-f 


-f 


It 


-t 


-t 


rf 


rf 


rf 


rf 


-+ 


rf 


rf 


rf 


rf 


rf 


CO 


CO 




o 


1—1 


































































>o 


■ — i 


CM 


o 


NO 


CO 


co 


1/5 


X 


CM 


r- 


rt 


O 


r~ 


it 


Ol 


r - 


rf 


, | 


00 


NO 


o. 


CM 


NO 


O 




o 


o 


LO 


CO 


ON 


lO 


CM 


o 


CO 


t^ 


NO 


tO 


-* 


■>* 


CO 


co 


CO 


01 


Ol 


Ol 




i — 1 


1 — i 


O 


O 


o 


O 


00 


CO 




o 


rf 


CO 


NO 


LO 


m 


in 


in 


if 


rf 


rt 


rt 


Tf 


■>+ 


rt 


rt 


rt 


Tf 


rt 


rf 


rf 


rf 


rf 


-* 


it 


^o 


co 


CO 


CO 




o 


r "" t 






























































NO 


l-H 


O 


-t 


CO 


rt 


o 


CO 


o 


01 


NO 


r-H 


r - 


m 


O 


r- 


IO 


CM 


Q> 


NO 


CO 


,— 


(0\ 


OJ 


NO 


^H 


rt 


CM 


o 


© 


01 


CO 


r~ 


01 


c* 


r~ 


NO 


rt 


co 


CO 


Ol 


CM 


<—< 


' — 1 


1 — l 


c 


O 


o 


o 


o 


O 


o 


00 


cc 


1 - 


r- 


NO 


o 


rf 


CC 


NO 


in 


m 


rf 


rt 


rf 


rt 


rt 


-f 


rf 


■<+ 


Tt 


-* 


Tf 


-+ 


-+ 


rf 


CO 


CO 


ro 


CO 


CO 


r^ 


CO 


co 


CO 




o 


^ 
























































Q. / 




























































/ 


1 — 1 


CM 


CO 


-f 


LO 


NO 


ir- 


cc- 


Q\ 


o 


P-H 


CM 


CO 


-+ 


to 


NO 


1 - 


M 


CTn 


O 


01 


rf 


S 


CO 


o 


O 


Q 


a 


i 


/ 








































CM 


CM 


(M 


CM 


rt 


rf 


5 


o 


/ C 





























































43 



STATISTICAL CHART 5 



Scores for Ranked Data 



The mean deviations of the 1 st , 2 nc % 3 f d . . . largest members of samples 
of different sizes; zero and negative values omitted. 



Ordinal 
number 




2 


3 


Size 
4 


of Sam 
5 


pie 
6 


7 


8 


9 


10 


1 




.56 


.85 


1.03 


1.16 


1.27 


1.35 


1.42 


1.49 


1.54 


2 








.30 


.50 


.64 


.76 


.85 


.93 


1.00 


3 












.20 


.35 


.47 


.57 


.66 


4 
















.15 


.27 


.38 


5 




















.12 




11 


12 


13 


14 


15 


16 


17 


18 


19 


20 


1 


1.59 


1.63 


1.67 


1.70 


1.74 


1.76 


1.79 


1.82 


1.84 


1.87 


2 


1.06 


1.12 


1.16 


1.21 


1.25 


1.28 


1.32 


1.35 


1.38 


1.41 


3 


.73 


.79 


.85 


.90 


.95 


.99 


1.03 


1.07 


1.10 


1.13 


4 


.46 


.54 


.60 


.66 


.71 


.76 


.81 


.85 


.89 


.92 


5 


.22 


.31 


.39 


.46 


.52 


.57 


.62 


.67 


.71 


.75 


6 




.10 


.19 


.27 


.34 


.39 


.45 


.50 


.55 


.59 


1-7 
t 








.09 


.17 


.23 


.30 


.35 


.40 


.45 


8 












.08 


.15 


.21 


.26 


.31 


9 
















.07 


.13 


.19 


10 




















.06 




21 


22 


23 


24 


25 


26 


27 


28 


29 


30 


1 


1.89 


1.91 


1.93 


1.95 


1.97 


1.98 


2.00 


2.01 


2.03 


2.04 


2 


1.43 


1.46 


1.48 


1.50 


1.52 


1.54 


1.56 


1.58 


1.60 


1.62 


3 


1.16 


1.19 


1.21 


1.24 


1.26 


1.29 


1.31 


1.33 


1.35 


1.36 


4 


.95 


.98 


1.01 


1.04 


1.07 


1.09 


1.11 


1.14 


1.16 


1.18 


5 


.78 


.82 


.85 


.88 


.91 


.93 


.96 


.98 


1.00 


1.03 


6 


.63 


.67 


.70 


.73 


.76 


.79 


.82 


.85 


.87 


.89 


7 


.49 


.53 


.57 


.60 


.64 


.67 


.70 


.73 


.75 


.78 


8 


.36 


.41 


.45 


.48 


.52 


.55 


.58 


.61 


.64 


.67 


9 


.24 


.29 


.33 


.37 


.41 


.44 


.48 


.51 


.54 


.57 


10 


.12 


.17 


.22 


.26 


.30 


.34 


.38 


.41 


.44 


.47 


11 




.06 


.11 


.16 


.20 


.24 


.28 


.32 


.35 


.38 


12 








.05 


.10 


.14 


.19 


.22 


.26 


.29 


13 












.05 


.09 


.13 


.17 


.21 


14 
















.04 


.09 


.12 


15 




















.04 



Tests of psychological preference and some other experimental data 
suffice to place a series of magnitudes in order of preference, without supplying 
metrical values. Analyses of variance, correlations, etc., can be carried out on 
such data by using the normal scores, appropriate to each position in order, in 
a sample of the size observed. Ties may be scored with the means of the ordinal 
values involved, but in such cases the sums of squares will require correction. 



44 



APPENDIX III 
NOTES ON INTRODUCTORY STATISTICS 1 

The purpose of these notes is to put in a written form the content of 
a 3-hour discussion on some basic statistical concepts, held at the Food 
Research Institute of the Department of Agriculture. 

The term statistics includes the numerical descriptions (figures) of 
the quantitative aspects of things, and the body of methods (subject) used 
for making decisions in the face of uncertainty. The decisions may vary. 
They may be concerned with estimating the probability of rain on a summer 
day after considering the available meteorological data, whether or not to 
accept a shipment of parts after only partial inspection, or how to set up an 
experimental plan to test several varieties of geese for tenderness. These 
notes deal with the concept of statistics as a body of methods. 

Statistics is helpful to research workers in many ways, such as by 
suggesting which to observe and how to observe it (Theory of Design of 
Experiments aid Theory of Sampling) and how to summarize results in forms 
that are comprehensible (Descriptive Statistics). Owing to experimental 
error (chance circumstance), data and predictions cannot be expected to 
agree exactly even if the scientist's theories are correct. Therefore, when 
mathematical results from the Theory of Chance are applied to statistical 
problems (Inferential Statistics), they help to draw conclusions from the 
data. 

SAMPLE AND POPULATION 

Sample (data, observations) is the "number" that have been observed. 
Population is the totality of all possible observations of the same kind. 

Sample results vary by chance, and the pattern of chance variation 
depends on the population from which the samples have been drawn. A 
sample is not a miniature replicate of the population, so when decisions 
about a population are based on a sample, it is necessary to make allowance 
for the role of chance. 

RANDOM SAMPLE 

If a sample is to be representative of the population, each member of 
the population should have an equal chance of being included in the sample. 
But this requirement is not enough in itself to make up a random sample. 



This appendix was prepared by Andres Petrasovits, Statistical Research Service, 
Canada Department of Agriculture, Ottawa. 



45 



A sample of size N is said to be random if each combination of N items in 
the population has an equal chance of being chosen. 

To stress the importance of the italicized words in the above definition, 
consider the following: 

Example: A college catalogue of students has 50 pages. A sample of 50 
students was taken by selecting at random one student from each page. 

Note that this is not a random sample, because a sample including two 
students listed on the same page has a zero chance of being chosen. 

It must be pointed out that samples that are not strictly random are 
often preferable to random samples on the grounds of convenience or of 
increased precision (Stratification). 

PARAMETER AND STATISTIC 

Samples are taken to learn about the population being sampled. A 
parameter is a quantitative characteristic of the population. A statistic 
is a mathematical function of the sample values or a value computed from 
the sample. 

Example 1 

Suppose we are interested in learning about the average weight of 
adult Canadians. The true average weight (call it pt) of Canadians is 
clearly impossible (physically and economically) to compute. It is possible, 
however, to take a sample of 100 Canadians, register their weights, and 
compute the mean of the sample weights. Let us denote this mean obtained 
from the sample X, and suppose that, in this case, we obtained X = 141.5 
pounds. As a first approximation it would be reasonable to regard 141.5 
pounds as an estimate of fi, the true but unknown average weight for all 
Canadians. 

Then we say: X is a statistic computed from the sample, |i is a 
parameter of the population, and X is an estimate of fi. 

Example 2 

If we are interested in the variability of weights, an indicator of 
dispersion that is often used is the range (largest minus smallest). As with 
the mean weights, the true range (call it R) of adult Canadian body weights 
cannot be practically obtained, but we can use the sample of size 100 
(from example 1) to obtain an estimate of the true range on the basis of the 
range observed in the sample. Let us denote the latter by r. 

Then we say: r is a statistic computed from the sample and R is a 
parameter of the population: r is an estimate of R. 



46 



Example 3 

A political poll is to be taken for Ontario to estimate the proportion 
of Liberals. Clearly, the true proportion of Liberals (call it 77) would be 
practically impossible to compute. We can, however, obtain the proportion 
of Liberals from the sample taken (call it p) and regard p as an estimate of 
77. Suppose that p was found to be 0.46, then we say that p = 0.46 is a 
statistic computed from the sample, 77 is a parameter of the population, 
and p is an estimate of 77. 

It should be pointed out that many statistics can be computed from a 
sample. In example 1, other statistics besides X may give an estimate of fi, 
for instance the most frequent value (mode) or the average of the extremes. 

POPULATION OF SAMPLE MEANS 

Consider the following finite population P: (1, 2, 3, 4). Note that two 
summary measures of interest about P are the population mean \l = 2.5 and 
the population range R = 4 - 1=3. 

Let us form all possible samples of size 2, say, without replacement 
from P and for each sample let us compute the sample mean. We have: 

Sample X 

(1, 2) 1.5 

(1, 3) 2 

(1, 4) 2.5 

(2, 3) 2.5 

(2, 4) 3 

(3, 4) 3.5 

The set of sample means 

Px = (1.5, 2, 2.5, 2.5, 3, 3.5) 
is called the population of sample means of size 2 obtained from the 
original population P. 



ORIGINAL POPULATION: P 



POPULATION OF SAMPLE MEANS: P 



:> 



1.5 2.5 
2 35 3 



When comparing the original population, P, with the population of sample 
means of size 2, P^, it should be noticed that: 

(i) the mean of the population of sample means is equal to the mean 
of the original population, 



47 



(ii) the variability (dispersion) in the population of sample means is 
smaller than the variability in the original population. 

The above results are illustrated by the previous example. 

For (i), 

(1 + 2 + 3 + 4)/4 = (1.5 + 2 + 2.5 + 2.5 + 3 + 3.5)/6 = 2.5, 

and for (ii), if we agree to use the range as a measure of variability, then 

range for original population = 4 - 1 = 3, and 

range for population of sample means = 3.5 - 1.5 = 2. 

As an exercise for the reader it may be useful to examine the population of 
sample means derived by taking all possible samples of size 3 from the 
population: 

P = (2, 3, 3, 7, 7, 0). 
NORMAL DISTRIBUTION 



Frequency Distribution 

The pattern (characteristics) of a population or sample is shown by 
grouping the data in a frequency table. This table is called the frequency 
distribution of the population or the sample. 

Example: The ages in years of 10 students in a classroom were: 
21, 20, 21, 22, 20, 22, 21, 20, 23, 21. 

Frequency distribution 



20 
21 

22 
23 



Frequency 
3 
4 

2 
1 



5 
>- 4- 

Z 7 

HI -■' 

S 2-1 

£ i 





GRAPHICAL REPRESENTION OF 
THE FREQUENCY DISTRIBUTION 



21 22 

AGE 



23 



Characterization of the Normal Distribution 

Roughly speaking, a normal distribution is a frequency distribution 
whose graphical description is "bell-shaped." It must be noted that not 
all bell-shaped distributions are normal. 

A normal distribution is fully determined by two parameters: the mean 
Qz) and the standard deviation (a). A full definition of the latter is available 
in any introductory statistics textbook. 



48 




The physical meaning of the 
quantities yt and a. 



INFLECTION 
POINT 



The mean, /x, establishes the "location" or "center" of the distribution. 
The standard deviation, a, measures dispersion and is given by the hori- 
zontal distance from xi to the inflection points of the normal curve. An 
important characteristic of the normal distribution is its symmetry about the 
mean xt. 




NORMAL CURVES WITH DIFFERENT MEANS (LOCATIONS) 
AND SAME STANDARD DEVIATION (DISPERSION) 



NORMAL CURVES WITH SAME MEAN (LOCATION) 
AND DIFFERENT STANDARD DEVIATIONS 



These figures show how the mean (/x) and the standard deviation (a) affect 
the appearance of the normal distribution. 

Example 

In a study of radioactivity, measurements and counts per minute were 
registered with their frequency. 



Frequency 


distribution 


Counts 




per 




minute 


Frequency 


9000 


3 


9100 


20 


9200 


31 


9300 


54 


9400 


81 


9500 


112 


9600 


88 


9700 


64 


9800 


37 


9900 


14 


10000 


6 



125 - 



100 " 



75 - 



? 50 * 



25 ' 



FREQUENCY DISTRIBUTION 



4 

nnn I 



9000 ' 9200 
9100 



9600 ' 9800 
9300 9500 9700 9900 

COUNTS PER MINUTE 



49 



CENTRAL LIMIT THEOREM 

The normal distribution arises from some natural populations (for 
example, radioactivity counts) and from all populations of sample means 
(of large enough sample size). This is an important fact in the Theory of 
Statistics and it is known as the Central Limit Theorem, which states that 
as the sample size increases, the distribution of the population of sample 
means becomes more like the "bell-shaped" normal distribution regardless 
of the shape of the distribution of the original population. 

Example: Consider the population: 

P l= (0, 1, 2, 3, 4, 5) 

and construct from it the population of sample means of size 2 (P2) and the 
population of sample means of size 4 (P 4 ). From the frequency distributions 
of P 1? P 2 , and P 4 construct graphical representations. It will be observed 
that the shape of the graphical representation of P 2 looks more bell-shaped 
than that of P l and that the shape of the graphical representation of P 4 
looks more bell-shaped than that of P 2 . 



Construction of P 2 


Construe 


tion of P 4 




Sample 


X 


Sample 


X 


Sample 


X 


Sample 


X 


(0, 1) 


0.5 


(1,4) 


2.5 


(0, 1, 2, 3) 


1.50 


(0, 2, 3, 5) 


2.50 


(0, 2) 


1.0 


(0, 5) 


3.0 


(0, 1, 2, 4) 


1.75 


(0, 2, 4, 5) 


2.75 


(0, 3) 


1.5 


(2, 3) 


2.5 


(0, 1, 2, 5) 


2.00 


(0, 3, 4, 5) 


3.00 


(0, 4) 


2.0 


(2, 4) 


3.0 


(0, 1, 3, 4) 


2.00 


(1, 2, 3, 4) 


2.50 


(0, 5) 


2.5 


(2, 5) 


3.5 


(0, 1, 3, 5) 


2.25 


(1, 2, 3, 5) 


2.75 


(1,2) 


1.5 


(3,4) 


3.5 


(0, 1, 4, 5) 


2.50 


(1, 2, 4, 5) 


3.00 


(1, 3) 


2.0 


(3, 5) 


4.0 


(0, 2, 3, 4) 


2.25 


(1, 3, 4, 5) 


3.25 






(4, 5) 


4.5 






(2, 3, 4, 5) 


3.50 



Frequency distribution of P l 



/alue 


Fre 


quency 







1 


1 




1 


2 




1 


3 




1 


4 




1 


5 




1 



12 3 4 

VALUE 



50 



Frequency distribution of P 2 
Sample 



mean 

0.5 
1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 



Frequency 

1 
1 
2 
2 
3 
2 
2 
1 
1 



f 

3- -I 




: 1 1 ! 


1 1 



0.5 



1.5 2.0 2.5 3.0 3.5 4.0 4.5 
SAMPLE MEAN 



Frequency distribution of P 4 
Sample 



mean 


Frequency 


1.50 


1 


1.75 


1 


2.00 


2 


2.25 


2 


2.50 


3 


2.75 


2 


3.00 


2 


3.25 


1 


3.50 


1 



u 



H 



1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 
SAMPLE MEAN 



The conclusions to be drawn from this illustration are: 

(i) As the sample size increases, the frequency distribution of the 
population of sample means approaches a normal distribution. 

(ii) As the sample size increases, the variability in the population of 
sample means diminishes. 

INFERENTIAL CONCEPTS 

Two inferential tools of great use in statistics are Tests of Hypothesis 
and Confidence Intervals. Although closelv related in their theoretical 
basis, they serve different needs from an applied viewpoint. 

Example: 

Suppose a scientist is investigating the number of bacteria per gram in 
frozen eggs. Suppose also that his main interest is in the mean number of 
bacteria per gram (call it /i). Two situations can be visualized: 



51. 



(i) The scientist may have some theory of his own or he may have 
other facts (another scientist's experiments) that lead him to 
believe that /i = 150. He may decide to use his observations to 
test if, in fact, fi = 150 is consistent with his experimental results. 

(ii) Or the scientist may want to estimate the mean number from his 
data and may not be concerned with checking any particular 
theory. Situation (i) leads to Statistical Tests of Hypothesis and 
situation (ii) leads to Confidence Intervals. 

PROBABILITY 

There are several interpretations of probability. Here we give the 
approach referred to as frequentist. In a long series of throws with a coin, 
heads and tails will occur approximately equally often. We then say that 
the probability of a head (or tail) turning up is %. In a long series of 
throws with a fair die, each of the six sides will occur in approximately 
1/6 of the total number of throws and the probability for any of the numbers 
is 1/6. Generally the concept of probability can be formulated by saying 
that in a very long series of trials any event will tend to occur with a 
relative frequency that is approximately equal to the probability of the 
event. 

TESTS OF HYPOTHESIS 

An example borrowed from criminal legal procedure may help to 
introduce some concepts. 

Null hypothesis: H : defendant is innocent. 

Alternative hypothesis: H a : defendant is guilty. 

The jury observes the evidence and reaches a verdict. The two types of 
error are: 

Type I error: an innocent person is found guilty. 

Type II error: a criminal is acquitted. 

The type I error consists of rejecting the null hypothesis when it is true. 
The type II error consists of accepting the null hypothesis when the 
alternative is true. 

An example of a statistical test of hypothesis is the following: A coin 
is known to be either fair (ht) or two headed (hh). After a single toss of the 
coin, a test of the hypothesis must be set up: 

H : the coin is fair 

versus H a : the coin is two headed. 



52 



The test of a statistical hypothesis consists of a decision rule to accept 
or reject the null hypothesis (H ) on the basis of relevant statistical 
information (outcome) of a single toss of the coin. There are many decision 
rules that can be used to construct a test. Some are better (more reasonable) 
than others. Here are three different decision rules, each of which can be 
used as a statistical test for the null hypothesis that the coin is ht versus 
the alternative that the coin is hh. 

D|: Toss the coin and if t shows up accept H ; if h shows up reject 

H - 
D 2 : Toss the coin and if t shows up accept H ; if h shows up accept 

H„- 

D 3 : Toss the coin and if it rains outside accept H ; if it does not 
rain outside reject H . 

If we must choose one of the decision rules as a basis for our test, D 3 , 
although it is a decision rule, can be disregarded on common sense grounds, 
since it does not use experimental evidence. However, when Dj and D 2 are 
compared, it is not clear which one is better. Therefore, a measure of the 
goodness of the decision rule that is to be used as a basis for a statistical 
test of hypothesis is needed. 

Associated with a decision rule are two quantities: a = P (making a 
type I error) which reads "probability of making a Type I error" and /3 = "P 
(making a type II error). Clearly, a desirable property for a statistical 
test is that both a and /3 be as low as possible. Thus, the pair (a, /3) 
provides a basis to measure the goodness of a statistical test of hypothesis. 
Unfortunately, the size of both a and /3 is not enough by itself to select 
the best one of several available tests of hypothesis. This point will be 
illustrated by computing a and /3 for decision rules D t and D 2 in the coin 
example. 

ForD^ 

a = P (making a type I error) 

= P( rejecting H when it is true) 

= P (h shows up with a fair coin) 



1 

= 2 



/3 = P (making a type II error) 

= P (accepting H when H a is true) 

= P (t shows up with a two-headed coin) 

= 

Therefore for Dj, a = A and /3 = 

For D 2 , 

a = P (making a type I error) 

= P (rejecting H when it is true) 

= 0, because according to D 2 we never reject H 



53 



/3 = P (making a type II error) 

= P (accepting H when H a is true) 
= P (heads or tails show up when coin is 2h) 

= 1, because heads will always appear, so the event (heads or tails) 
will always occur. 



Therefore: 




a 
D 2 



1 



The above table shows that the decision rule (D j or D 2 ) that should be 
used cannot be chosen only by considering the probabilities a and /3 that 
correspond to each decision rule because their relative importance is 
unknown. Consideration of the economic importance of the type I and type II 
errors is necessary when choosing which decision rule to use for a parti- 
cular situation. Further discussion of this point is beyond the scope of 
these notes. 

The probability of making a type I error is called the level of signifi- 
cance of the test, while the quantity, 1 minus P (type II error), is referred 
to as the power of the test. In statistical methodology, if no economic 
criteria are available to weigh the importance of the type I and type II 
errors, the usual way to select a test is to fix the level of significance 
arbitrarily, say 0.05, to consider only the decision rules for which the level 
of significance is 0.05, and then to choose the decision rule for which the 
power is maximum. 

CONFIDENCE INTERVALS 

Research workers often have to estimate parameters. An estimate of a 
parameter given by the corresponding statistic is unlikely to be precisely 
equal to the parameter, so it is necessary to show the margin of variability 
to which it is subject. A way to do this is to specify an interval within 
which we may be "confident" that the parameter lies. Such an interval is 
called a confidence interval. 

Example: A sample of 10 adult Canadians is taken to estimate the mean 
Canadian adult weight (fi). 



POPULATION 


SAMPLE 




"\ 198 192 




\ 178 164 


^SAMPLE 


\_1 ^ 158 190 

' ) 200 200 




/ 165 160 



54 



If it is assumed that Canadian adult weights are normally distributed, 
by using a technique available in any textbook on statistics, the 95% 
confidence interval for n can be computed from the sample and is 160.75, 
190.25. The meaning of a confidence interval is often misunderstood. It 
does not mean that the probability is 0.95 that the population mean (//) 
lies between 160.75 and 190.25. It means that if many random samples 
of size 10 are taken from the population of adult Canadian weights and for 
each sample a 95% confidence interval for the mean is computed according 
to the above technique, then, in a very long series 95% of the confidence 
intervals so computed will contain the true mean. 



REFERENCES 

1. Amerine, M. A., and C. S. Ough. 1964. The Sensory Evaluation of 

Californian Wines. Lab. Pract. Vol. 13, No. 8. 

2. Baker, R. A. 1964. Taste and Odour in Water. Lab. Pract. Vol. 13, 

No. 8. 

3. Bengtsson, K., and E. Helm. 1953. Principles of Taste Testing. 

Wallerstein Laboratories Communications. 

4. Caul, J. F. 1957. The Profile Method of Flavor Analysis. Adv. in Food 

Research, 7: 1. 

5. Cochran, W.G., and G.M. Cox. 1957. Experimental Designs. John Wiley 

and Sons, New York, N.Y. 

6. Dawson, E. H., J.L. Brogdon, and S. McManus. 1963. Sensory Testing 

of Differences in Taste. Food Technol. Vol. 17, No. 9. 

7. Duncan, D. B. 1955. Multiple Range and Multiple F Tests. Biometrics, 

Vol. II. 

8. Eindhoven, J., D. Peryam, F. Heiligman, and G. A. Baker. 1964. 

Effect of Sample Sequence on Food Preference. J. Food Sci. 
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9. Fisher, R. A., and F. Yates. 1942. Statistical Tables. Oliver and 

Boyd Ltd., Edinburgh and London. 

10. Kramer, A., J. Cooler, M. Modery, and B.A. Twigg. 1963. Numbers of 

Tasters Required to Determine Consumer Preference for Fruit 
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11. Lowe, B. 1963. Experimental Cookery, John Wiley and Sons. New York, 

N.Y. 

12. Pangborn, R. M. V. 1964. Sensory Evaluation of Food at the University 

of California. Lab. Pract. Vol. 13, No. 7. 



55 



13. Pettit, L. A. 1958. Informational Bias in Flavor Preference Testing. 

Food Technol. Vol. 12, No. 1. 

14. Peryam, D. R. 1964. Sensory Testing at the Quartermaster Food and 

Container Institute. Lab. Pract. Vol. 13, No. 7. 

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ment of Taste Quality in the Confectionery Industry. Biometrics, 
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Experiment Station Paper No. 56. 

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Judgments in a Test on Flavor Evaluations for Preference. Food 
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ence Panels and Trained Panel Scores on Dry Whole Milk. J. Dairy 
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20. Scheffe, H. 1952. An Analysis of Variance for Paired Comparisons. 

J.Am. Statist. Ass. 47: 381-400. 

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of the Flavor Profile. Food Technol. Vol. 11, p. 20. 

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Iowa State College, Ames. 

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Report. Food Technol. Vol. 16, No. 3. 

24. Tilgner, D. J. 1962. Dilution Tests for Odor and Flavor Analysis. 

Food Technol. Vol. 16, No. 2. 



ADDITIONAL SOURCES OF INFORMATION 

Bartlett, F. 1964. The Evaluation of Sensory Experiences. Lab. Pract. 
Vol. 13, No. 7. 

Christie, E. M. 1964. Taste Testing in the C.S.I. R.O. Lab. Pract. Vol. 13, 

No. 7. 

Committee on Sensory Evaluation of the Institute of Food Technology. 1964. 
Sensory Testing Guide for Panel Evaluation of Food and 
Beverages. 



56 



Ellis, B. H. 1961. A Guide Book for Sensory Testing. Continental Can. 
Co. Inc. 

Gridgeman, N. T. 1959. Pair Comparisons, With and Without Ties. 
Biometrics, 15: 382-388. 

Gridgeman, N. T. 1959. Sensory Item Sorting. Biometrics, 15: 298—306. 

Gridgeman, N. T. 1961. A Comparison of Some Taste Test Methods. J. Food 
Sci. 26: 171-177. 

Harper, R. 1964. The Sensory Evaluation of Food and Drink. An Overview. 
Lab. Pract. Vol. 13, No. 7. 

Harries, J. M. 1964. Sensory Testing at the Ministry of Agriculture, Fish- 
eries and Food. Lab. Pract. Vol. 13, No. 7. 

Kramer, A., and B. Twigg. 1962. Fundamentals of Quality Control in the 
Food Industry. The AVI Publishing Co. Inc., Westport, Connecti- 
cut. 

McCowen, P. 1964. Sensory Testing at Lyons Ltd. Lab. Pract. Vol. 13, 

No. 8. 

Sullivan, F., and J. F. Caul. 1964. Applications of Flavor Profile to Food 
and Beverage Packaging Problems. Lab. Pract. Vol. 13, No. 7. 

Tilgner, D. J. 1964. Sensory Analysis at the Politichnika Gdanska. Lab. 
Pract. Vol. 13, No. 7. 

Wallis, W. A., and H. V. Roberts. 1956. Statistics, New Approach. The Free 
Press, Glencoe, Chicago, Illinois. 



57 



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